Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G)...
Transcript of Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G)...
![Page 1: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/1.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenability of operator algebras on Banachspaces, I
Volker Runde
University of Alberta
NBFAS, Leeds, June 1, 2010
![Page 2: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/2.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Philosophical musings
A quote
Big things are rarely amenable!
N.N., Istanbul, 2004.
Thus. . .
AMENABLE ≈ SMALL
![Page 3: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/3.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Philosophical musings
A quote
Big things are rarely amenable!
N.N., Istanbul, 2004.
Thus. . .
AMENABLE ≈ SMALL
![Page 4: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/4.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Philosophical musings
A quote
Big things are rarely amenable!
N.N., Istanbul, 2004.
Thus. . .
AMENABLE ≈ SMALL
![Page 5: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/5.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Philosophical musings
A quote
Big things are rarely amenable!
N.N.,
Istanbul, 2004.
Thus. . .
AMENABLE ≈ SMALL
![Page 6: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/6.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Philosophical musings
A quote
Big things are rarely amenable!
N.N., Istanbul, 2004.
Thus. . .
AMENABLE ≈ SMALL
![Page 7: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/7.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Philosophical musings
A quote
Big things are rarely amenable!
N.N., Istanbul, 2004.
Thus. . .
AMENABLE ≈ SMALL
![Page 8: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/8.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Philosophical musings
A quote
Big things are rarely amenable!
N.N., Istanbul, 2004.
Thus. . .
AMENABLE
≈ SMALL
![Page 9: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/9.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Philosophical musings
A quote
Big things are rarely amenable!
N.N., Istanbul, 2004.
Thus. . .
AMENABLE ≈ SMALL
![Page 10: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/10.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Philosophical musings
A quote
Big things are fucking rarely amenable!
N.N., Istanbul, 2004.
Thus. . .
AMENABLE ≈ SMALL
![Page 11: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/11.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G ) which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 12: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/12.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G ) which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 13: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/13.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group.
A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G ) which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 14: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/14.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean
on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G ) which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 15: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/15.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean on L∞(G )
is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G ) which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 16: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/16.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗
such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G ) which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 17: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/17.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G ) which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 18: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/18.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G ) which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 19: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/19.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable
if there is a mean on L∞(G ) which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 20: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/20.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G )
which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 21: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/21.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G ) which is leftinvariant,
i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 22: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/22.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G ) which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 23: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/23.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Mean things
Definition
Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.
Definition (J. von Neumann 1929; M. M. Day, 1949)
G is amenable if there is a mean on L∞(G ) which is leftinvariant, i.e.,
〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),
where(Lxφ)(y) := φ(xy) (y ∈ G ).
![Page 24: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/24.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 25: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/25.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 26: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/26.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable:
M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 27: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/27.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 28: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/28.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable:
use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 29: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/29.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 30: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/30.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 31: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/31.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable
and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 32: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/32.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G ,
then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 33: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/33.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 34: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/34.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable
and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 35: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/35.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G ,
then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 36: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/36.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 37: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/37.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G
and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 38: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/38.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G
are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 39: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/39.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,
then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 40: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/40.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 41: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/41.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union
of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 42: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/42.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G
such that G =⋃α Hα, then G is amenable.
![Page 43: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/43.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα,
then G is amenable.
![Page 44: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/44.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some amenable groups. . .
Examples
1 Compact groups amenable: M = Haar measure.
2 Abelian groups are amenable: use Markov–Kakutani to getM.
Really nice hereditary properties!
1 If G is amenable and H < G , then H is amenable.
2 If G is is amenable and N C G , then G/N is amenable.
3 If G and N C G are such that N and G/N are amenable,then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroupsof G such that G =
⋃α Hα, then G is amenable.
![Page 45: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/45.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, I
Example
Let F2 be the free group in two generators.Assume that there is a left invariant mean M on `∞(F2).Define
µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.
Then
µ is finitely additive,
µ(F2) = 1, and
µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).
![Page 46: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/46.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, I
Example
Let F2 be the free group in two generators.Assume that there is a left invariant mean M on `∞(F2).Define
µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.
Then
µ is finitely additive,
µ(F2) = 1, and
µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).
![Page 47: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/47.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, I
Example
Let F2 be the free group in two generators.
Assume that there is a left invariant mean M on `∞(F2).Define
µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.
Then
µ is finitely additive,
µ(F2) = 1, and
µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).
![Page 48: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/48.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, I
Example
Let F2 be the free group in two generators.Assume that there is a left invariant mean M on `∞(F2).
Defineµ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.
Then
µ is finitely additive,
µ(F2) = 1, and
µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).
![Page 49: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/49.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, I
Example
Let F2 be the free group in two generators.Assume that there is a left invariant mean M on `∞(F2).Define
µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.
Then
µ is finitely additive,
µ(F2) = 1, and
µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).
![Page 50: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/50.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, I
Example
Let F2 be the free group in two generators.Assume that there is a left invariant mean M on `∞(F2).Define
µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.
Then
µ is finitely additive,
µ(F2) = 1, and
µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).
![Page 51: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/51.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, I
Example
Let F2 be the free group in two generators.Assume that there is a left invariant mean M on `∞(F2).Define
µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.
Then
µ is finitely additive,
µ(F2) = 1, and
µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).
![Page 52: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/52.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, I
Example
Let F2 be the free group in two generators.Assume that there is a left invariant mean M on `∞(F2).Define
µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.
Then
µ is finitely additive,
µ(F2) = 1,
and
µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).
![Page 53: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/53.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, I
Example
Let F2 be the free group in two generators.Assume that there is a left invariant mean M on `∞(F2).Define
µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.
Then
µ is finitely additive,
µ(F2) = 1, and
µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).
![Page 54: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/54.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, I
Example
Let F2 be the free group in two generators.Assume that there is a left invariant mean M on `∞(F2).Define
µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.
Then
µ is finitely additive,
µ(F2) = 1, and
µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).
![Page 55: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/55.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, II
Example (continued. . . )
For x ∈ {a, b, a−1, b−1} set
W (x) := {w ∈ F2 : w starts with x}.
Let w ∈ F2 \W (a). Then a−1w ∈W (a−1), therefore
w ∈ aW (a−1),
and thusF2 = W (a) ∪ aW (a−1).
Similarly,F2 = W (b) ∪ bW (b−1)
holds.
![Page 56: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/56.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, II
Example (continued. . . )
For x ∈ {a, b, a−1, b−1} set
W (x) := {w ∈ F2 : w starts with x}.
Let w ∈ F2 \W (a). Then a−1w ∈W (a−1), therefore
w ∈ aW (a−1),
and thusF2 = W (a) ∪ aW (a−1).
Similarly,F2 = W (b) ∪ bW (b−1)
holds.
![Page 57: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/57.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, II
Example (continued. . . )
For x ∈ {a, b, a−1, b−1}
set
W (x) := {w ∈ F2 : w starts with x}.
Let w ∈ F2 \W (a). Then a−1w ∈W (a−1), therefore
w ∈ aW (a−1),
and thusF2 = W (a) ∪ aW (a−1).
Similarly,F2 = W (b) ∪ bW (b−1)
holds.
![Page 58: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/58.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, II
Example (continued. . . )
For x ∈ {a, b, a−1, b−1} set
W (x) := {w ∈ F2 : w starts with x}.
Let w ∈ F2 \W (a). Then a−1w ∈W (a−1), therefore
w ∈ aW (a−1),
and thusF2 = W (a) ∪ aW (a−1).
Similarly,F2 = W (b) ∪ bW (b−1)
holds.
![Page 59: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/59.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, II
Example (continued. . . )
For x ∈ {a, b, a−1, b−1} set
W (x) := {w ∈ F2 : w starts with x}.
Let w ∈ F2 \W (a).
Then a−1w ∈W (a−1), therefore
w ∈ aW (a−1),
and thusF2 = W (a) ∪ aW (a−1).
Similarly,F2 = W (b) ∪ bW (b−1)
holds.
![Page 60: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/60.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, II
Example (continued. . . )
For x ∈ {a, b, a−1, b−1} set
W (x) := {w ∈ F2 : w starts with x}.
Let w ∈ F2 \W (a). Then a−1w ∈W (a−1),
therefore
w ∈ aW (a−1),
and thusF2 = W (a) ∪ aW (a−1).
Similarly,F2 = W (b) ∪ bW (b−1)
holds.
![Page 61: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/61.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, II
Example (continued. . . )
For x ∈ {a, b, a−1, b−1} set
W (x) := {w ∈ F2 : w starts with x}.
Let w ∈ F2 \W (a). Then a−1w ∈W (a−1), therefore
w ∈ aW (a−1),
and thusF2 = W (a) ∪ aW (a−1).
Similarly,F2 = W (b) ∪ bW (b−1)
holds.
![Page 62: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/62.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, II
Example (continued. . . )
For x ∈ {a, b, a−1, b−1} set
W (x) := {w ∈ F2 : w starts with x}.
Let w ∈ F2 \W (a). Then a−1w ∈W (a−1), therefore
w ∈ aW (a−1),
and thusF2 = W (a) ∪ aW (a−1).
Similarly,F2 = W (b) ∪ bW (b−1)
holds.
![Page 63: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/63.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, II
Example (continued. . . )
For x ∈ {a, b, a−1, b−1} set
W (x) := {w ∈ F2 : w starts with x}.
Let w ∈ F2 \W (a). Then a−1w ∈W (a−1), therefore
w ∈ aW (a−1),
and thusF2 = W (a) ∪ aW (a−1).
Similarly,F2 = W (b) ∪ bW (b−1)
holds.
![Page 64: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/64.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, III
Example (continued even further)
Since
F2 = {ε}·∪W (a)
·∪W (a−1)
·∪W (b)
·∪W (b−1),
we have
1 = µ(F2)
≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))
≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))
= µ(F2) + µ(F2)
= 2,
which is nonsense.
![Page 65: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/65.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, III
Example (continued even further)
Since
F2 = {ε}·∪W (a)
·∪W (a−1)
·∪W (b)
·∪W (b−1),
we have
1 = µ(F2)
≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))
≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))
= µ(F2) + µ(F2)
= 2,
which is nonsense.
![Page 66: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/66.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, III
Example (continued even further)
Since
F2 = {ε}·∪W (a)
·∪W (a−1)
·∪W (b)
·∪W (b−1),
we have
1 = µ(F2)
≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))
≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))
= µ(F2) + µ(F2)
= 2,
which is nonsense.
![Page 67: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/67.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, III
Example (continued even further)
Since
F2 = {ε}·∪W (a)
·∪W (a−1)
·∪W (b)
·∪W (b−1),
we have
1 = µ(F2)
≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))
≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))
= µ(F2) + µ(F2)
= 2,
which is nonsense.
![Page 68: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/68.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, III
Example (continued even further)
Since
F2 = {ε}·∪W (a)
·∪W (a−1)
·∪W (b)
·∪W (b−1),
we have
1 = µ(F2)
≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))
≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))
= µ(F2) + µ(F2)
= 2,
which is nonsense.
![Page 69: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/69.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, III
Example (continued even further)
Since
F2 = {ε}·∪W (a)
·∪W (a−1)
·∪W (b)
·∪W (b−1),
we have
1 = µ(F2)
≥ µ(W (a)) + µ(
a
W (a−1)) + µ(W (b)) + µ(
b
W (b−1))
≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))
= µ(F2) + µ(F2)
= 2,
which is nonsense.
![Page 70: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/70.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, III
Example (continued even further)
Since
F2 = {ε}·∪W (a)
·∪W (a−1)
·∪W (b)
·∪W (b−1),
we have
1 = µ(F2)
≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))
≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))
= µ(F2) + µ(F2)
= 2,
which is nonsense.
![Page 71: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/71.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, III
Example (continued even further)
Since
F2 = {ε}·∪W (a)
·∪W (a−1)
·∪W (b)
·∪W (b−1),
we have
1 = µ(F2)
≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))
≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))
= µ(F2) + µ(F2)
= 2,
which is nonsense.
![Page 72: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/72.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, III
Example (continued even further)
Since
F2 = {ε}·∪W (a)
·∪W (a−1)
·∪W (b)
·∪W (b−1),
we have
1 = µ(F2)
≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))
≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))
= µ(F2) + µ(F2)
= 2,
which is nonsense.
![Page 73: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/73.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, III
Example (continued even further)
Since
F2 = {ε}·∪W (a)
·∪W (a−1)
·∪W (b)
·∪W (b−1),
we have
1 = µ(F2)
≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))
≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))
= µ(F2) + µ(F2)
= 2,
which is nonsense.
![Page 74: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/74.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
. . . and a non-amenable one, III
Example (continued even further)
Since
F2 = {ε}·∪W (a)
·∪W (a−1)
·∪W (b)
·∪W (b−1),
we have
1 = µ(F2)
≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))
≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))
= µ(F2) + µ(F2)
= 2,
which is nonsense.
![Page 75: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/75.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 76: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/76.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 77: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/77.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable
or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 78: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/78.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 79: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/79.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 80: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/80.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 81: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/81.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup,
e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 82: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/82.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 83: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/83.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R)
with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 84: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/84.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 85: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/85.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2,
or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 86: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/86.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 87: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/87.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R)
with N ≥ 3 equipped with the discretetopology.
![Page 88: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/88.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3
equipped with the discretetopology.
![Page 89: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/89.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Consequences
Hence, G is amenable if. . .
1 G is solvable or
2 G is locally finite.
But G is not amenable if. . .
G contains F2 as closed subgroup, e.g., if
G = SL(N,R) with N ≥ 2,
G = GL(N,R) with N ≥ 2, or
G = SO(N,R) with N ≥ 3 equipped with the discretetopology.
![Page 90: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/90.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras via approximatediagonals
Definition (B. E. Johnson, 1972)
A Banach algebra A is said to be amenable if it possesses anapproximate diagonal, i.e., a bounded net (dα)α in theprojective tensor product A⊗A such that
a · dα − dα · a→ 0 (a ∈ A)
anda∆dα → a (a ∈ A)
with ∆ : A⊗A→ A denoting multiplication.
![Page 91: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/91.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras via approximatediagonals
Definition (B. E. Johnson, 1972)
A Banach algebra A is said to be amenable if it possesses anapproximate diagonal, i.e., a bounded net (dα)α in theprojective tensor product A⊗A such that
a · dα − dα · a→ 0 (a ∈ A)
anda∆dα → a (a ∈ A)
with ∆ : A⊗A→ A denoting multiplication.
![Page 92: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/92.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras via approximatediagonals
Definition (B. E. Johnson, 1972)
A Banach algebra A is said to be amenable
if it possesses anapproximate diagonal, i.e., a bounded net (dα)α in theprojective tensor product A⊗A such that
a · dα − dα · a→ 0 (a ∈ A)
anda∆dα → a (a ∈ A)
with ∆ : A⊗A→ A denoting multiplication.
![Page 93: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/93.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras via approximatediagonals
Definition (B. E. Johnson, 1972)
A Banach algebra A is said to be amenable if it possesses anapproximate diagonal,
i.e., a bounded net (dα)α in theprojective tensor product A⊗A such that
a · dα − dα · a→ 0 (a ∈ A)
anda∆dα → a (a ∈ A)
with ∆ : A⊗A→ A denoting multiplication.
![Page 94: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/94.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras via approximatediagonals
Definition (B. E. Johnson, 1972)
A Banach algebra A is said to be amenable if it possesses anapproximate diagonal, i.e., a bounded net (dα)α in theprojective tensor product A⊗A
such that
a · dα − dα · a→ 0 (a ∈ A)
anda∆dα → a (a ∈ A)
with ∆ : A⊗A→ A denoting multiplication.
![Page 95: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/95.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras via approximatediagonals
Definition (B. E. Johnson, 1972)
A Banach algebra A is said to be amenable if it possesses anapproximate diagonal, i.e., a bounded net (dα)α in theprojective tensor product A⊗A such that
a · dα − dα · a→ 0 (a ∈ A)
anda∆dα → a (a ∈ A)
with ∆ : A⊗A→ A denoting multiplication.
![Page 96: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/96.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras via approximatediagonals
Definition (B. E. Johnson, 1972)
A Banach algebra A is said to be amenable if it possesses anapproximate diagonal, i.e., a bounded net (dα)α in theprojective tensor product A⊗A such that
a · dα − dα · a→ 0 (a ∈ A)
anda∆dα → a (a ∈ A)
with ∆ : A⊗A→ A denoting multiplication.
![Page 97: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/97.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras via approximatediagonals
Definition (B. E. Johnson, 1972)
A Banach algebra A is said to be amenable if it possesses anapproximate diagonal, i.e., a bounded net (dα)α in theprojective tensor product A⊗A such that
a · dα − dα · a→ 0 (a ∈ A)
anda∆dα → a (a ∈ A)
with ∆ : A⊗A→ A denoting multiplication.
![Page 98: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/98.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras and amenable groups
Theorem (B. E. Johnson, 1972)
The following are equivalent for a locally compact group G :
1 G is amenable;
2 L1(G ) is amenable.
Grand theme
Let C be a class of Banach algebras. Characterize the amenablemembers of C!
![Page 99: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/99.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras and amenable groups
Theorem (B. E. Johnson, 1972)
The following are equivalent for a locally compact group G :
1 G is amenable;
2 L1(G ) is amenable.
Grand theme
Let C be a class of Banach algebras. Characterize the amenablemembers of C!
![Page 100: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/100.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras and amenable groups
Theorem (B. E. Johnson, 1972)
The following are equivalent for a locally compact group G :
1 G is amenable;
2 L1(G ) is amenable.
Grand theme
Let C be a class of Banach algebras. Characterize the amenablemembers of C!
![Page 101: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/101.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras and amenable groups
Theorem (B. E. Johnson, 1972)
The following are equivalent for a locally compact group G :
1 G is amenable;
2 L1(G ) is amenable.
Grand theme
Let C be a class of Banach algebras. Characterize the amenablemembers of C!
![Page 102: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/102.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras and amenable groups
Theorem (B. E. Johnson, 1972)
The following are equivalent for a locally compact group G :
1 G is amenable;
2 L1(G ) is amenable.
Grand theme
Let C be a class of Banach algebras. Characterize the amenablemembers of C!
![Page 103: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/103.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras and amenable groups
Theorem (B. E. Johnson, 1972)
The following are equivalent for a locally compact group G :
1 G is amenable;
2 L1(G ) is amenable.
Grand theme
Let C be a class of Banach algebras. Characterize the amenablemembers of C!
![Page 104: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/104.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras and amenable groups
Theorem (B. E. Johnson, 1972)
The following are equivalent for a locally compact group G :
1 G is amenable;
2 L1(G ) is amenable.
Grand theme
Let C be a class of Banach algebras.
Characterize the amenablemembers of C!
![Page 105: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/105.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Amenable Banach algebras and amenable groups
Theorem (B. E. Johnson, 1972)
The following are equivalent for a locally compact group G :
1 G is amenable;
2 L1(G ) is amenable.
Grand theme
Let C be a class of Banach algebras. Characterize the amenablemembers of C!
![Page 106: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/106.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
More from abstract harmonic analysis
Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)
The following are equivalent:
1 M(G ) is amenable;
2 G is amenable and discrete.
Theorem (B. E. Forrest & VR, 2005)
The following are equivalent:
1 A(G ) is amenable;
2 G has an abelian subgroup of finite index.
![Page 107: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/107.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
More from abstract harmonic analysis
Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)
The following are equivalent:
1 M(G ) is amenable;
2 G is amenable and discrete.
Theorem (B. E. Forrest & VR, 2005)
The following are equivalent:
1 A(G ) is amenable;
2 G has an abelian subgroup of finite index.
![Page 108: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/108.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
More from abstract harmonic analysis
Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)
The following are equivalent:
1 M(G ) is amenable;
2 G is amenable and discrete.
Theorem (B. E. Forrest & VR, 2005)
The following are equivalent:
1 A(G ) is amenable;
2 G has an abelian subgroup of finite index.
![Page 109: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/109.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
More from abstract harmonic analysis
Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)
The following are equivalent:
1 M(G ) is amenable;
2 G is amenable and discrete.
Theorem (B. E. Forrest & VR, 2005)
The following are equivalent:
1 A(G ) is amenable;
2 G has an abelian subgroup of finite index.
![Page 110: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/110.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
More from abstract harmonic analysis
Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)
The following are equivalent:
1 M(G ) is amenable;
2 G is amenable
and discrete.
Theorem (B. E. Forrest & VR, 2005)
The following are equivalent:
1 A(G ) is amenable;
2 G has an abelian subgroup of finite index.
![Page 111: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/111.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
More from abstract harmonic analysis
Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)
The following are equivalent:
1 M(G ) is amenable;
2 G is amenable and discrete.
Theorem (B. E. Forrest & VR, 2005)
The following are equivalent:
1 A(G ) is amenable;
2 G has an abelian subgroup of finite index.
![Page 112: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/112.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
More from abstract harmonic analysis
Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)
The following are equivalent:
1 M(G ) is amenable;
2 G is amenable and discrete.
Theorem (B. E. Forrest & VR, 2005)
The following are equivalent:
1 A(G ) is amenable;
2 G has an abelian subgroup of finite index.
![Page 113: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/113.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
More from abstract harmonic analysis
Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)
The following are equivalent:
1 M(G ) is amenable;
2 G is amenable and discrete.
Theorem (B. E. Forrest & VR, 2005)
The following are equivalent:
1 A(G ) is amenable;
2 G has an abelian subgroup of finite index.
![Page 114: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/114.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
More from abstract harmonic analysis
Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)
The following are equivalent:
1 M(G ) is amenable;
2 G is amenable and discrete.
Theorem (B. E. Forrest & VR, 2005)
The following are equivalent:
1 A(G ) is amenable;
2 G has an abelian subgroup of finite index.
![Page 115: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/115.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
More from abstract harmonic analysis
Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)
The following are equivalent:
1 M(G ) is amenable;
2 G is amenable and discrete.
Theorem (B. E. Forrest & VR, 2005)
The following are equivalent:
1 A(G ) is amenable;
2 G has an abelian subgroup
of finite index.
![Page 116: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/116.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
More from abstract harmonic analysis
Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)
The following are equivalent:
1 M(G ) is amenable;
2 G is amenable and discrete.
Theorem (B. E. Forrest & VR, 2005)
The following are equivalent:
1 A(G ) is amenable;
2 G has an abelian subgroup of finite index.
![Page 117: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/117.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 118: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/118.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 119: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/119.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable
and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 120: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/120.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range,
then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 121: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/121.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable.
Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 122: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/122.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular,
if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 123: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/123.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A,
then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 124: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/124.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 125: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/125.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that
both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 126: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/126.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I
and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 127: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/127.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I
are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 128: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/128.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable,
then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 129: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/129.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 130: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/130.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable
and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 131: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/131.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A,
then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 132: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/132.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 133: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/133.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;
2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 134: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/134.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;
3 I is weakly complemented in A, i.e., I⊥ is complementedin A∗.
![Page 135: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/135.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A,
i.e., I⊥ is complementedin A∗.
![Page 136: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/136.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From old to new. . .
Hereditary properties
1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.
2 If I C A is such that both I and A/I are amenable, then A
is amenable.
3 If A is amenable and I C A, then the following areequivalent:
1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented
in A∗.
![Page 137: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/137.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 138: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/138.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 139: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/139.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 140: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/140.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable
for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 141: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/141.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 142: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/142.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 143: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/143.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 144: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/144.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed,
but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 145: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/145.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjoint
subalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 146: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/146.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 147: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/147.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 148: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/148.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 149: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/149.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 150: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/150.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Grand theme, reprise!
The “meaning” of amenability
What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?
1 all C ∗-algebras;
2 all von Neumann algebras;
3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);
4 all algebras K(E );
5 all algebras B(E ).
Remember. . .
AMENABLE ≈ SMALL
![Page 151: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/151.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 152: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/152.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N.
Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 153: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/153.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear,
we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 154: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/154.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification,
i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 155: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/155.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 156: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/156.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 157: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/157.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive
ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 158: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/158.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive
for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 159: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/159.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 160: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/160.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 161: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/161.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 162: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/162.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive,
but not completely positive.
![Page 163: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/163.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Complete positivity
If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,
T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).
Definition
T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.
Example
M2 → M2, a 7→ at
is positive, but not completely positive.
![Page 164: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/164.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear C ∗-algebras
Definition
A C ∗-algebra A is called nuclear if there are nets (nλ)λ ofpositive integers and of completely positive contractions
Mnλ
Aid
-
T λ
-
A
Sλ
-
such that(Sλ ◦ Tλ)a→ a (a ∈ A).
![Page 165: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/165.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear C ∗-algebras
Definition
A C ∗-algebra A is called nuclear if there are nets (nλ)λ ofpositive integers and of completely positive contractions
Mnλ
Aid
-
T λ
-
A
Sλ
-
such that(Sλ ◦ Tλ)a→ a (a ∈ A).
![Page 166: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/166.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear C ∗-algebras
Definition
A C ∗-algebra A is called nuclear
if there are nets (nλ)λ ofpositive integers and of completely positive contractions
Mnλ
Aid
-
T λ
-
A
Sλ
-
such that(Sλ ◦ Tλ)a→ a (a ∈ A).
![Page 167: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/167.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear C ∗-algebras
Definition
A C ∗-algebra A is called nuclear if there are nets (nλ)λ ofpositive integers
and of completely positive contractions
Mnλ
Aid
-
T λ
-
A
Sλ
-
such that(Sλ ◦ Tλ)a→ a (a ∈ A).
![Page 168: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/168.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear C ∗-algebras
Definition
A C ∗-algebra A is called nuclear if there are nets (nλ)λ ofpositive integers and of completely positive contractions
Mnλ
Aid
-
T λ
-
A
Sλ
-
such that(Sλ ◦ Tλ)a→ a (a ∈ A).
![Page 169: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/169.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear C ∗-algebras
Definition
A C ∗-algebra A is called nuclear if there are nets (nλ)λ ofpositive integers and of completely positive contractions
Mnλ
Aid
-
T λ
-
A
Sλ
-
such that(Sλ ◦ Tλ)a→ a (a ∈ A).
![Page 170: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/170.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear C ∗-algebras
Definition
A C ∗-algebra A is called nuclear if there are nets (nλ)λ ofpositive integers and of completely positive contractions
Mnλ
Aid
-
T λ
-
A
Sλ
-
such that(Sλ ◦ Tλ)a→ a (a ∈ A).
![Page 171: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/171.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear C ∗-algebras
Definition
A C ∗-algebra A is called nuclear if there are nets (nλ)λ ofpositive integers and of completely positive contractions
Mnλ
Aid
-
T λ
-
A
Sλ
-
such that(Sλ ◦ Tλ)a→ a (a ∈ A).
![Page 172: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/172.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclearity and amenability
Theorem (A. Connes, 1978)
A C ∗-algebra A is nuclear if it is amenable and A∗ is separable.
Theorem (U. Haagerup, 1983)
All nuclear C ∗-algebras are amenable.
Theorem (A. Connes, U. Haagerup, et al.)
The following are equivalent for a C ∗-algebra A:
1 A is nuclear;
2 A is amenable.
![Page 173: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/173.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclearity and amenability
Theorem (A. Connes, 1978)
A C ∗-algebra A is nuclear if it is amenable and A∗ is separable.
Theorem (U. Haagerup, 1983)
All nuclear C ∗-algebras are amenable.
Theorem (A. Connes, U. Haagerup, et al.)
The following are equivalent for a C ∗-algebra A:
1 A is nuclear;
2 A is amenable.
![Page 174: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/174.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclearity and amenability
Theorem (A. Connes, 1978)
A C ∗-algebra A is nuclear if it is amenable
and A∗ is separable.
Theorem (U. Haagerup, 1983)
All nuclear C ∗-algebras are amenable.
Theorem (A. Connes, U. Haagerup, et al.)
The following are equivalent for a C ∗-algebra A:
1 A is nuclear;
2 A is amenable.
![Page 175: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/175.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclearity and amenability
Theorem (A. Connes, 1978)
A C ∗-algebra A is nuclear if it is amenable and A∗ is separable.
Theorem (U. Haagerup, 1983)
All nuclear C ∗-algebras are amenable.
Theorem (A. Connes, U. Haagerup, et al.)
The following are equivalent for a C ∗-algebra A:
1 A is nuclear;
2 A is amenable.
![Page 176: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/176.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclearity and amenability
Theorem (A. Connes, 1978)
A C ∗-algebra A is nuclear if it is amenable and A∗ is separable.
Theorem (U. Haagerup, 1983)
All nuclear C ∗-algebras are amenable.
Theorem (A. Connes, U. Haagerup, et al.)
The following are equivalent for a C ∗-algebra A:
1 A is nuclear;
2 A is amenable.
![Page 177: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/177.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclearity and amenability
Theorem (A. Connes, 1978)
A C ∗-algebra A is nuclear if it is amenable and A∗ is separable.
Theorem (U. Haagerup, 1983)
All nuclear C ∗-algebras are amenable.
Theorem (A. Connes, U. Haagerup, et al.)
The following are equivalent for a C ∗-algebra A:
1 A is nuclear;
2 A is amenable.
![Page 178: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/178.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclearity and amenability
Theorem (A. Connes, 1978)
A C ∗-algebra A is nuclear if it is amenable and A∗ is separable.
Theorem (U. Haagerup, 1983)
All nuclear C ∗-algebras are amenable.
Theorem (A. Connes, U. Haagerup, et al.)
The following are equivalent for a C ∗-algebra A:
1 A is nuclear;
2 A is amenable.
![Page 179: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/179.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclearity and amenability
Theorem (A. Connes, 1978)
A C ∗-algebra A is nuclear if it is amenable and A∗ is separable.
Theorem (U. Haagerup, 1983)
All nuclear C ∗-algebras are amenable.
Theorem (A. Connes, U. Haagerup, et al.)
The following are equivalent for a C ∗-algebra A:
1 A is nuclear;
2 A is amenable.
![Page 180: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/180.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclearity and amenability
Theorem (A. Connes, 1978)
A C ∗-algebra A is nuclear if it is amenable and A∗ is separable.
Theorem (U. Haagerup, 1983)
All nuclear C ∗-algebras are amenable.
Theorem (A. Connes, U. Haagerup, et al.)
The following are equivalent for a C ∗-algebra A:
1 A is nuclear;
2 A is amenable.
![Page 181: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/181.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclearity and amenability
Theorem (A. Connes, 1978)
A C ∗-algebra A is nuclear if it is amenable and A∗ is separable.
Theorem (U. Haagerup, 1983)
All nuclear C ∗-algebras are amenable.
Theorem (A. Connes, U. Haagerup, et al.)
The following are equivalent for a C ∗-algebra A:
1 A is nuclear;
2 A is amenable.
![Page 182: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/182.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclearity and amenability
Theorem (A. Connes, 1978)
A C ∗-algebra A is nuclear if it is amenable and A∗ is separable.
Theorem (U. Haagerup, 1983)
All nuclear C ∗-algebras are amenable.
Theorem (A. Connes, U. Haagerup, et al.)
The following are equivalent for a C ∗-algebra A:
1 A is nuclear;
2 A is amenable.
![Page 183: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/183.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear von Neumann algebras
Theorem (S. Wasserman, 1976)
The following are equivalent for a von Neumann algebra M:
1 M is nuclear;
2 M is subhomogeneous, i.e.,
M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)
with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.
Corollary
B(H) is amenable if and only if dim H <∞.
![Page 184: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/184.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear von Neumann algebras
Theorem (S. Wasserman, 1976)
The following are equivalent for a von Neumann algebra M:
1 M is nuclear;
2 M is subhomogeneous, i.e.,
M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)
with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.
Corollary
B(H) is amenable if and only if dim H <∞.
![Page 185: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/185.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear von Neumann algebras
Theorem (S. Wasserman, 1976)
The following are equivalent for a von Neumann algebra M:
1 M is nuclear;
2 M is subhomogeneous, i.e.,
M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)
with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.
Corollary
B(H) is amenable if and only if dim H <∞.
![Page 186: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/186.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear von Neumann algebras
Theorem (S. Wasserman, 1976)
The following are equivalent for a von Neumann algebra M:
1 M is nuclear;
2 M is subhomogeneous, i.e.,
M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)
with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.
Corollary
B(H) is amenable if and only if dim H <∞.
![Page 187: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/187.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear von Neumann algebras
Theorem (S. Wasserman, 1976)
The following are equivalent for a von Neumann algebra M:
1 M is nuclear;
2 M is subhomogeneous,
i.e.,
M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)
with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.
Corollary
B(H) is amenable if and only if dim H <∞.
![Page 188: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/188.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear von Neumann algebras
Theorem (S. Wasserman, 1976)
The following are equivalent for a von Neumann algebra M:
1 M is nuclear;
2 M is subhomogeneous, i.e.,
M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)
with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.
Corollary
B(H) is amenable if and only if dim H <∞.
![Page 189: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/189.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear von Neumann algebras
Theorem (S. Wasserman, 1976)
The following are equivalent for a von Neumann algebra M:
1 M is nuclear;
2 M is subhomogeneous, i.e.,
M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)
with n1, . . . , nk ∈ N
and M1, . . . ,Mk abelian.
Corollary
B(H) is amenable if and only if dim H <∞.
![Page 190: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/190.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear von Neumann algebras
Theorem (S. Wasserman, 1976)
The following are equivalent for a von Neumann algebra M:
1 M is nuclear;
2 M is subhomogeneous, i.e.,
M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)
with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.
Corollary
B(H) is amenable if and only if dim H <∞.
![Page 191: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/191.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear von Neumann algebras
Theorem (S. Wasserman, 1976)
The following are equivalent for a von Neumann algebra M:
1 M is nuclear;
2 M is subhomogeneous, i.e.,
M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)
with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.
Corollary
B(H) is amenable if and only if dim H <∞.
![Page 192: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/192.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear von Neumann algebras
Theorem (S. Wasserman, 1976)
The following are equivalent for a von Neumann algebra M:
1 M is nuclear;
2 M is subhomogeneous, i.e.,
M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)
with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.
Corollary
B(H) is amenable
if and only if dim H <∞.
![Page 193: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/193.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Nuclear von Neumann algebras
Theorem (S. Wasserman, 1976)
The following are equivalent for a von Neumann algebra M:
1 M is nuclear;
2 M is subhomogeneous, i.e.,
M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)
with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.
Corollary
B(H) is amenable if and only if dim H <∞.
![Page 194: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/194.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 195: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/195.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 196: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/196.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G
on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 197: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/197.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H
is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 198: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/198.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π
from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 199: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/199.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G
into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 200: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/200.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H)
which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 201: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/201.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G
and the operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 202: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/202.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the weak operator topology on B(H).
We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 203: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/203.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).
We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 204: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/204.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 205: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/205.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).We call π:
1 unitary
if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 206: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/206.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 207: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/207.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable
if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 208: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/208.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation,
i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 209: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/209.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H)
such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 210: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/210.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary,
and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 211: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/211.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded
if
supg∈G‖π(g)‖ <∞.
![Page 212: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/212.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Representations of locally compact groups
Definition
A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).We call π:
1 unitary if π(G ) consists of unitaries,
2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and
3 uniformly bounded if
supg∈G‖π(g)‖ <∞.
![Page 213: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/213.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 214: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/214.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary
=⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 215: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/215.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable
=⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 216: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/216.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 217: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/217.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 218: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/218.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable.
Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 219: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/219.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G
is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 220: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/220.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 221: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/221.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 222: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/222.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold,
i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 223: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/223.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G
such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 224: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/224.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable
already amenable?
Fact
It’s false for F2!
![Page 225: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/225.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 226: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/226.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 227: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/227.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Unitarizability and amenability
Obvious. . .
π unitary =⇒ π unitarizable =⇒ π uniformly bounded.
Theorem (J. Dixmier, 1950)
Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.
Big open question
Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?
Fact
It’s false for F2!
![Page 228: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/228.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 229: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/229.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G ,
we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 230: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/230.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π
and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 231: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/231.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 232: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/232.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 233: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/233.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 234: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/234.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable,
then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 235: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/235.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 236: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/236.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 237: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/237.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable,
then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 238: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/238.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable,
so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 239: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/239.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H)
such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 240: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/240.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
From groups to algebras
Integration of representations
If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:
π(f ) :=
∫G
f (g)π(g) dg (f ∈ L1(G )).
Easy
If G is amenable, then A := π(L1(G )) is amenable.
Slightly more difficult
If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).
![Page 241: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/241.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?
![Page 242: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/242.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?
![Page 243: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/243.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed,
but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?
![Page 244: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/244.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed, but not necessarily self-adjoint
subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?
![Page 245: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/245.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed, but not necessarily self-adjoint subalgebra of B(H)
iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?
![Page 246: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/246.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra
if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?
![Page 247: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/247.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)
such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?
![Page 248: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/248.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?
![Page 249: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/249.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?
![Page 250: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/250.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H)
similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?
![Page 251: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/251.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H)
(which is necessarily nuclear)?
![Page 252: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/252.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
The similarity problem for amenable operatoralgebras
Definition
A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).
Big open question
Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?
![Page 253: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/253.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 254: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/254.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 255: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/255.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra
of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 256: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/256.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).
Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 257: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/257.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 258: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/258.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 259: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/259.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1.
We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 260: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/260.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable
if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 261: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/261.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 262: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/262.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 263: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/263.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed,
1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 264: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/264.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable
subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 265: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/265.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H).
Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 266: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/266.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 267: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/267.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?
![Page 268: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/268.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative,
even generated by one operator?
![Page 269: Amenability of operator algebras on Banach spaces, I · G isamenable if there is a mean on L1(G) which isleft invariant, i.e., hL x˚;Mi= h˚;Mi (x 2G;˚2L1(G)); where (L x˚)(y)](https://reader034.fdocuments.in/reader034/viewer/2022043010/5fa0d0b9df173f02f85dacce/html5/thumbnails/269.jpg)
Amenability ofoperator
algebras onBanachspaces, I
Volker Runde
Prelude
Amenablegroups
AmenableBanachalgebras
C∗- and vonNeumannalgebras
Similarityproblems
Some partial results
Theorem (J. A. Gifford, <2006)
Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).
Definition
Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .
Theorem (D. Blecher & C. LeMerdy, 2004)
Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.
Open question
What if A is commutative, even generated by one operator?