# Noncommutative Geometry in Hawai’imath.hawaii.edu/~dellaiera/NCG_Seminar.pdf · De nition 1.3.1....

### Transcript of Noncommutative Geometry in Hawai’imath.hawaii.edu/~dellaiera/NCG_Seminar.pdf · De nition 1.3.1....

Noncommutative Geometry in Hawai’i

Clement Dell’Aiera

These are the notes I took from the talks of the Noncommutative Geometry, co-organizedby Erik Guentner, Rufus Willett and myself from Fall 2017 up until now (Spring 2019).There are probably mistakes and typos.

Schedule, abstracts for the seminar (and some posts) are available on my personnalwebpage.

2

Contents

1 C∗-simplicity 51.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 How to prove C∗-simplicity? . . . . . . . . . . . . . . . . . . . . . . 71.1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Completely positive maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Injective C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Description of commutative injective algebras . . . . . . . . . . . . 111.4 Furstenburg boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Dynamical characterization of C∗-simplicity . . . . . . . . . . . . . . . . . 141.6 Another proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.7 Thompson’s group V is C∗-simple . . . . . . . . . . . . . . . . . . . . . . . 19

2 Weakly and non-weakly band dominated operators 232.1 Approximation of band dominated operators . . . . . . . . . . . . . . . . . 232.2 Approximation by bounded operators . . . . . . . . . . . . . . . . . . . . . 232.3 Weakly band dominated operators . . . . . . . . . . . . . . . . . . . . . . . 242.4 Characterizing membership in the Roe algebra . . . . . . . . . . . . . . . . 25

2.4.1 Roe’s question on conditions (2) and (4) . . . . . . . . . . . . . . . 262.5 Heart of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.6 Property (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Noncommutative geometry 393.1 Basic objects and constructions . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Quantum groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Why SUq(2)? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 TQFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.2 Summary of the talks . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Reminder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Dynamical Property (T) 474.1 Kazdhan projections and failure of K-exactness . . . . . . . . . . . . . . . 53

5 Cartan subalgebras 555.1 Groupoids of germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Generalized Gelfand transform . . . . . . . . . . . . . . . . . . . . . . . . . 565.3 Roe algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6 Classification and the UCT 59

3

4

Chapter 1

C∗-simplicity

1.1 General introduction

Let Γ be a discrete group. We will recall two equivalence relations on the set(?) of unitaryrepresentations of Γ, which are group homomorphisms

π : Γ→ U(Hπ)

where U(Hπ) stands for the unitary group of a complex Hilbert space Hπ. We will referto such a representation as (π,Hπ) or even just π or Hπ if no confusion is possible.

Let π and σ be two representations of Γ.

• π w σ iff there exists a unitary u : Hπ → Hσ such that

uπγu∗ = σγ ∀γ ∈ Γ.

• π ≈ σ iff there exists a sequence of unitaries un : Hπ → Hσ such that

‖unπγu∗n − σγ‖ → 0 ∀γ ∈ Γ.

Fact: It turns out that for a lot of groups (e.g. finite, abelian, compact, simple Liegroups,...), these two notions coincide

π ≈ σ iff π w σ for π, σ irreducible.

Let Γ be the collection of all representations of Γ. A very hard problem is to describe

Γ/ ≈ .

It can be done sometimes, e.g. for Z the irreducible representations are given by thecircle, and any representation decomposes more or less uniquely into these.

Let us recall that the (left) regular representation

λ : Γ→ U(l2Γ)

is defined by λg(δh) = δgh. The reduced C∗-algebra C∗r (Γ) is the closure under the operatornorm of the image of the regular representation, i.e.

C∗r (Γ) = spanλγγ∈Γ.

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Chapter 1. C∗-simplicity

A representation π is tempered if it extends to a ∗-representation of C∗r (Γ). This happensiff the linear extension

π : C[Γ]→ B(Hπ)

satisfies ‖π(a)‖ ≤ ‖λ(a)‖,∀a ∈ C[Γ].

Fact: All representations are tempered iff the group is amenable.

Another (very hard) problem is to describe

Γr/ ≈ .

Definition 1.1.1. Γ is C∗-simple if C∗r (Γ) is simple, i.e. admits no proper two sidedclosed ideal.

Theorem 1.1.2 (Voiculescu). Γ is C∗-simple iff Γr/ ≈ is a point.

Examples of C∗-simple groups:

• Non abelian free groups;

• Torsion free hyperbolic groups;

• PSL(n,Z);

• Thompson’s group V .

Non C∗-simple examples

Recall that a group Γ if the trivial representation

1Γ : Γ→ U(C) = S1; γ 7→ id = 1;

is tempered. As a consequence, non trivial amenable groups are not C∗-simple as 1 ≈ λ(dim(l2Γ) 6= 1).

More generally if there exists an amenable normal subgroup K C Γ, then the quasi regularrepresentation

λΓ/K : Γ→ U(l2(Γ/K)); λΓ/K(γ)(δxK) = δγxK ;

is tempered, hence if K is not trivial, Γ is not C∗-simple. In particular any semi-directproduct K oH with K amenable and non trivial is not C∗-simple.

Amenability being stable by extensions and increasing unions, any group has a largestnormal amenable subgroup R C Γ called the amenable radical. The previous discussionshows that if Γ is C∗-simple, thenR = e. The converse does not hold and was completelyanswered by Kennedy et al.

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1.1. General introduction

1.1.1 How to prove C∗-simplicity?

a la Powers [10].

Definition 1.1.3. A group Γ is a Powers group if for every finite subset F ⊂ Γ thereexists a partition

Γ = C∐

D

and a finite number of elements γ1, ..., γn ∈ Γ with

• γC ∩ C = ∅ for every γinF :

• γiD ∩ γjD = ∅ for every i 6= j.

Examples:

• The free group on two generators F2 (Powers [10]);

• Many other examples using ”North-South” type dynamics (De la Harpe, Bridson,Osin).

Let us write a few words about the technique Powers used. For F2 = 〈a, b〉, let

τ : C∗r (F2)→ C; a 7→ 〈δe, aδe〉

be the canonical tracial state.

Theorem 1.1.4 (Powers [10]). For every a ∈ C∗r (Γ),

τ(x) = lim1

mn

∑i=1,nj=1,m

λiaλjbxλ

−jb λ−ia

Corollary 1.1.5. F2 is C∗-simple.

Proof. Let J C C∗r (F2) be an ideal. For x ∈ C∗r (F2) let xmn =∑

i=1,nj=1,m λiaλ

jbxλ

−jb λ−ia .

If x ∈ J then (x∗x)mn ∈ J so τ(x∗x)1C∗r (F2) ∈ J‖ ‖

. If J is not trivial, it contains a nonzero element x, which forces 1C∗r (F2) ∈ J as τ(x∗x) > 0. This ensures that J = C∗r (F2)and we are done.

Corollary 1.1.6. C∗r (F2) has a unique tracial state.

Proof. Let τ ′ be a tracial state on C∗r (F2). Then for x ∈ C∗r (F2),

τ ′(x) = τ ′(xmn)→ τ ′(τ(x)1) = τ(x)τ ′(1) = τ(x).

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Chapter 1. C∗-simplicity

1.1.2 Definitions

We only consider discrete countable groups, usually denoted by Γ.

Definition 1.1.7. A group is said to be C∗-simple if its reduced C∗-algebra is simple, i.e.has no proper closed two sided ideals.

A motivation for the interest toward such a notion can be the following result of Murrayand Von Neumman: the Von Neumman algebra L(Γ) is simple (no proper weakly closedtwo sided ideals) iff it is a factor iff Γ is ICC (infinite conjugacy classes, i.e. all non trivialconjugacy classes are infinite). Another one is that simplicity is one out of the 5 criteria(unital simple separable UCT with finite nuclear dimension) needed in the classificationtheorem obtained by Winter et. al.

Recall that, given two unitary representations of Γ, we say that π is weakly contained inσ and write

π < σ

if every positive type function associated to π can be approximated uniformly on compactsets by finite sums of such things associated to σ. In other words, if for every ξ ∈ Hπ,F ⊆ Γ finite and every ε > 0, there exists η1, η2, ..., ηk such that

|〈π(s)ξ, ξ〉 −∑i

〈σ(s)ηi, ηi〉| < ε ∀s ∈ F.

Remark: one can restricts to convex combinations of normalized positive type functions.

If π < σ, then the identity C[Γ]→ C[Γ] extends to a surjective ∗-morphisms

C∗σ(Γ)→ C∗π(Γ).

Indeed, it suffices to show that for every a ∈ C[Γ],

‖π(a)‖ ≤ ‖σ(a)‖.

As ‖π(a)‖2 = ‖π(a∗a)‖, we can suppose a positive. Then

〈π(s)ξ, ξ〉 ≤∑i

ti〈σ(s)ηi, ηi〉+ ε

≤ ‖σ(a)‖+ ε

hence ‖π(a)‖ ≤ ‖σ(a)‖+ ε, and let just ε go to zero.

Definition 1.1.8. A group Γ is C∗-simple if its reduced C∗-algebra is simple (i.e. has noproper closed two sided ideal).

Theorem 1.1.9. If Γ has a non trivial amenable normal subgroup, then it is not C∗-simple.

Proof. Let N be a normal amenable subgroup of Γ. Let (Fk) be a sequence of Folnersets for N , and

ξk =1

|Fk|χFk∈ l2(Γ)

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1.2. Completely positive maps

Then

〈λΓ(s)ξk, ξk〉 = 2− |Fk∆sFk||Fk|

wich is 0 if s /∈ N , and goes to 1 as n goes to infinity if s ∈ N . In other words

〈λΓ(s)ξk, ξk〉 → 〈λΓ/N(s)δeN , δeN〉,

which shows that λΓ/N < λΓ. This gives us a surjective ∗-morphism

φ : C∗r (Γ)→ C∗Γ/N(Γ).

A faster way which still works out when the ambient group is only locally compact is topoint out that, N being amenable,

1N < λN ,

ensures by inductionIndΓ

N1N = λΓ/N < IndΓNλN = λΓ.

But if n ∈ N is non trivial, λΓ(n) is non trivial and sent to λΓ/N(n) = 1 via φ, so thatKer φ is a proper ideal in C∗r (Γ).

After the talk, Erik Guentner suggested the following proof. It is even shorter anddoesn’t assume any knowledge about weak containment or induction of representations.It is a weakening of the following fact: when Γ is amenable, the trivial representation1Γ : C∗max(Γ)→ C extends to the reduced C∗-algebra.

Indeed let a ∈ C[Γ] and (Fn) be a sequence of Folner sets for the support of a. Defineξn = 1

|Fn|12χFn ∈ l2(Γ). Then, suppose a is positive, and compute

〈aξn, ξn〉 =∑

s∈ supp a

aγ|Fn ∩ sFn||Fn|

→ ‖a‖1Γ

so that ‖a‖r ≤ ‖a‖1Γ.

Now if N is a normal amenable subgroup of Γ...

We saw that F2 is C∗-simple, yet it has a copy of Z as an amenable subgroup (non normal),and a normal (non amenable) subgroup: the commutator subgroup, which is an infiniterank free group, 〈[x, y] : x, y ∈ F2〉 = F([an, bm];n,m). Both conditions are necessary.

This result led to following (false) conjecture: a group is C∗-simple iff it has no non trivialamenable normal subgroups.

1.2 Completely positive maps

If A and B are C∗-algebra, then a linear map φ : A→ B is called completely positive if

φ(n)(a) = (φ(aij))ij ≥ 0 ∀a ∈Mn(A)+.

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Chapter 1. C∗-simplicity

Denote CP (A,B) the normed vector space of completely positive maps form A to B.S(A) denotes the state space of A, endowed with the weak-∗ topology (it’s then a convexsubspace of A∗, compact when A is unital).

Then:

• CP (C(X), C(Y )) ∼= C(Y, P (X)) via µy(f) = Φ(f)(y);

• CP (A,C(Y )) ∼= C(Y, S(A)) via ωy(f) = Φ(a)(y);

• What about CP (C(X), B)? Continuous sections on the continuous field of C∗-algebras

⊕ω∈S(B) B(Hω).

1.3 Injective C∗-algebras

Recall that an abelian group M is injective if, given any injective homomorphism ofabelian group A → B, any homomorphism A→M extends to a homomorphism B →M .In words: any homomorphism into M extends to super-objects. We will often use thefollowing commutative diagram

B

A M

∃

to represent this situation. We will now turn to a analog notion in the C∗-algebraic world.

Definition 1.3.1. A C∗-algebra M is injective if, given an inclusion of C∗-algebra A ⊂ B,any injective ∗-homomorphism A→M extends to B by a contractive completely positive(CCP) map.

B

A M

∃ ccp

Even if the straight arrow are here supposed to be ∗-homomorphism, Stinespring’s dilationtheorem ensures that we can suppose all the arrows to be only CCP maps. We will saythat M is Γ-injective if Γ acts by automorphisms on all the C∗-algebras in the diagram,and all the arrows are Γ-equivariant.

We will define a particular class of compact spaces acted upon by Γ, called Γ-boundaries,and show that there exists a maximal Γ-boundary ∂FΓ, called the Thurston boundary.

The first major goal of this presentation is to show that C(∂FΓ) is Γ-injective.

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1.4. Furstenburg boundary

1.3.1 Description of commutative injective algebras

Lemma 1.3.2. If M is injective and S ⊂M , define

AnnM(S) = m ∈M | sm = 0∀s ∈ S.

Then there exists a projection p ∈M satisfying AnnM(S) = pM .

Proof. This is true if M = B(H) for some Hilbert space. In the general case, embed Munitally in some B(H). By injectivity of M , there exists a CCP map E : B(H) → Msuch that E(m) = m,∀m ∈ M so M ⊂ dom(E) (multiplicative domain). There exists aprojection p ∈ B(H) with AnnB(H)(S) = pB(H) (take the projection on ∩s∈SKer(s)). Ifs ∈ S,

sE(p) = E(sp) = 0 hence E(p) ∈ AnnM(S).

Moreover if m ∈ AnnM(S) ⊂ pB(H), pm = m and

E(p)m = E(pm) = E(m) = m

so that for m = E(p), we get E(p) is a projection. This also proves that E(p)AnnM(S) =AnnM(S). A slight fiddling ensures then that AnnM(S) = E(p)M .

Corollary 1.3.3. Let X be a compact Hausdorff space. If C(X) is injective then X isStonean, i.e. U is open for every open subset U ⊂ X.

Proof. Let U ⊂ X be open, and S = C0(U). By the previous lemma, there exists aprojection p ∈ C(X) such that AnnC(X)(S) = pC(X). But p cannot be anyone else than

the characteristic function of Uc

so that 1− p = χU is continuous and U is open.

Note: Infinite compact Stonean spaces are not metrizable (not even second countable).Suppose the contrary and get a sequence xi → x in X and open sets Un = B(xn, εn), withεn such that Un ∩ Um = ∅ for every n 6= m. Set U = ∪nU2n, then x ∈ U (U is open) soxn ∈ U for large n but xn /∈ U for n odd.

1.4 Furstenburg boundary

If Γ is a discrete group acting on a compact Hausdorff space X (we will just say that X isa Γ-space), the space of probability measures Prob(X) endowed with the weak-∗ topologyis homeomorphic to the state space S(C(X)) with the topology of simple convergence.We identify X with a closed subspace of Prob(X) with the help of the Dirac masses (X → Prob(X);x 7→ δx is an embedding). Recall that the action can be extended toProb(X), which is then a Γ-space by Banach-Alaoglu’s theorem.

Definition 1.4.1. A Γ-space X is:

• minimal if the only Γ-invariant closed subset of X are itself and ∅;

• strongly proximal if Γ.µweak−∗

contains δx for some x ∈ X;

• a Γ-boundary if it is minimal and strongly proximal

X ⊂ Γ.µweak−∗ ∀µ ∈ Prob(X).

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Chapter 1. C∗-simplicity

Example: Let SL(2,Z) act on the projective line RP 1 (the quotient of R2\0 by thegroup of dilations) given by (

a bc d

)(xy

)=

(ax+ bycx+ dy

).

Most g ∈ Sl(2,Z) are acting hyperbolically (two distinct eigenspaces, one expansive onecontractive). Take µ ∈ Prob(RP 1), and a generic element g ∈ SL(2,Z). As n goes to ∞,

gn.µ→weak−∗ δExpanding eigenspace

unless µ(contractive eigenspace) > 0, hence

δExpanding eigenspaceg∈SL(2,Z) ⊂ Γ.µwk−∗

.

Exercise: the set of these is dense in RP 1 ⊂ Prob(RP 1).

Theorem 1.4.2 (Furstenburg). There exists a Γ-boundary ∂FΓ (now called the Fursten-burg boundary) such that for any Γ-boundary X there exists a continuous Γ-equivariantsurjection ∂FΓ X.

Proof. Let B be the class of all Γ-boundaries. It is non empty as it contains the pointspace. Take

Z =∏Y ∈B

Y

which is compact by Tychonoff’s theorem. Equip Z by the diagonal Γ-action.

• It is strongly proximal: for any µ ∈ Prob(Z), a diagonal argument gives a weak-∗convergent net gi.µ→ δz for some z ∈ Z.

• It is not minimal, but Zorn’s lemma ensures the existence of a minimal closed Γ-invariant subset ∂FΓ of Z.

We obtain the desired map as the composition of the inclusion ∂FΓ → Z with the pro-jection on the X-factor Z X.

Theorem 1.4.3 (Kalantar-Kennedy). C(∂FΓ) is Γ-injective.

Lemma 1.4.4. There exists a bijective correspondence between the completely positivemaps from C(X) to C(Y ) and the continous maps from Y to Prob(X). The statementremains true if one asks for equivariance. send to a previous section on CP maps

Lemma 1.4.5 (Furstenburg). Let X and Y be two Γ-boundaries. Then any Γ-equivariantmapX → Prob(Y ) has image in Y , i.e. any UCP map C(X)→ C(Y ) is a ∗-homomorphism!Moreover there is at most one such map.

Proof. Take µ : X → Prob(Y ). The image µ(X) ⊂ Prob(Y ) is a closed Γ-invariantsubspace: by strong proximality of Y , there exists y ∈ Y such that

δy ∈ Γ.µxwk−∗ ⊂ µ(X).

By minimality of Y , Γ.µxwk−∗∩Y = Y , By minimality of X, µ−1(Y ) = X i.e. µ(X) ⊂ Y .

Let µ, η : X → Prob(Y ) be two such maps. Then 12µ + 1

2η, µ and η all take values in Y

so that they are all equal.

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1.4. Furstenburg boundary

Corollary 1.4.6. Any equivariant UCP map C(∂FΓ)→ C(∂FΓ) is the identity.

Recall that if A is a unital Γ-algebra, its state space S(A) is convex compact Γ-space.

Proposition 1.4.7 (Gleason). Let Z ⊂ S(A) be a Γ-invariant closed convex subspace,which is minimal w.r.t. these properties. (Such a thing exists by Zorn’s lemma.) Then

∂exZ = φ ∈ Z | φ is not a non trivial convex combination of anything in Z

is a Γ-boundary.

• •

•

Figure 1.1: Two examples with Z in blue and ∂exZ in black.

Proof. There is a barycenter map β : Prob(Z)→ Z such that∫Z

fdµ = f(β(µ)) ∀f ∈ C(Z) affine.

Indeed, if µ = δz, β(µ) = z and if µ =∑αiδzi with 0 ≤ αi ≤ 1 and

∑αi = 1, then

β(µ) =∑αizi. Finite convex combinations are weak-∗ dense in Prob(Z) by the Hahn-

Banach separation theorem. As β is weak-∗ continuous, and affine so uniformly weak-∗continuous, it extends to the whole space Prob(Z).

Note: β is Γ-equivariant continuous and satisfies β(µ) = z ∈ ∂exZ iff µ = δz.

Then, for any µ ∈ Prob(Z),

β(conv(Γµ)) = conv(Γβ(µ)) = Z,

the first equality coming from continuity, Γ-equivariance and affinity. Now, ∂exZ is min-imal, and if µ ∈ ∂exZ, then

AFINIR

We are now ready for the main result of this section.

Theorem 1.4.8 (Kalantar-Kennedy). C(∂FΓ) is Γ-injective.

Proof. First, observe that l∞(Γ) is Γ-injective. Let indeed A ⊂ B be an inclusion ofC∗-algebras and φ : A → l∞(Γ) a ∗-homomorphism. Then eve φ is a state on A, so itextends to a state Ψ on B. Define φ : B → l∞(Γ) by

φ(b)(γ) = Ψ(γ−1.b).

13

Chapter 1. C∗-simplicity

Then Ψ is a UCP Γ-equivariant map that extends φ.

Now, producing ucp equivariant maps

C(∂FΓ) l∞(Γ) C(∂FΓ)α β

is sufficient to conclude, as their composition must be the identity by corollary 1.4.6.

Define α : C(∂FΓ)→ l∞(Γ) by fixing µ ∈ Prob(∂FΓ) and set

α(f)(γ) = µ(γ−1.f).

By Gleason’s theorem 1.4.7, there is a Γ-boundary X ⊂ S(l∞(Γ)). By universal propertyof ∂FΓ, we have an equivariant surjection ∂FΓ X ⊂ S(l∞(Γ)). By duality, we get aΓ-equivariant ucp map

Ψ : l∞(Γ)→ C(∂FΓ)

and we are done.

As a final remark, one can point out that this last proof used the following useful fact: ifB is injective and φ : A → B is a split injective Γ-ucp map, then A is injective. We usethis with A = C(∂FΓ) and B = l∞(Γ).

1.5 Dynamical characterization of C∗-simplicity

(Facts we are using:C(∂FΓ) is Γ-injective, in particular any Γ-equivariant u.c.p. C(∂fΓ)→ A is split, so is anisometric embedding,∂FΓ is totally disconnected, )

The goal of this section is to prove the following theorem.

Theorem 1.5.1. Γ is C∗-simple iff the action of Γ on ∂FΓ is free.

Let’s do first the forward direction.

Suppose the action is free. First, to show C∗r (Γ) is simple, it is enough to show that anyrepresentation

π : C∗r (Γ)→ B(H)

is injective.

By Arveson’s extension theorem, π extends to a u.c.p. map

φ : C(∂FΓ)or Γ→ B(H).

Its restriction φ0 to C(∂FΓ) is Γ-equivariant, because C(∂FΓ) is in the multiplicative do-main of φ0, and thus must be an isometric embedding, by Γ-injectivity of C(∂FΓ) (it issplit because C ⊆ B(H)). The equivariant u.c.p. map φ0 is an isomorphism onto itsimage: extend its inverse form im φ0 to im φ and denote the resulting u.c.p map by τ .

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1.5. Dynamical characterization of C∗-simplicity

Claim: Ψ = τ φ is the canonical expectation E : C(∂FΓ) or Γ → C(∂FΓ) which isfaithful. This implies π is injective.

Let’s end up with the claim.

• Ψ|C(∂F Γ) = idC(∂F Γ). Indeed, τ is the inverse of φ0 = φC(∂F Γ).

• If γ 6= eΓ, the action being free, for every x there exists a function f ∈ C(∂FΓ) suchthat

f(x) 6= 0 and f(s−1x) = 0.

Now C(∂FΓ) is in the multiplicative domain of Ψ, so

Ψ(λs)f = Ψ(λsf) = Ψ((sf)λs) = (sf)Ψ(λs)

which evaluated at x gives Ψ(λs)(x) = 0, for all x, so Ψ(λs) = 0.

The other direction is more intricated. It consists in two steps:

1. if x ∈ ∂FΓ, then the stabilizer Γx is amenable, which implies that λΓ/Γx < λΓ,

2. if X is a X is a Γ-boundary, and γ 6= 0 such that int(Xs) 6= ∅, then λΓ ≮ λΓ/Γx , sothat the kernel of C∗r (Γ)→ C∗λΓ/Γx

(Γ) is a non trivial two sided closed ideal.

This, together with the fact that ∂FΓ is topologically free iff it is free, concludes the proof.

First bullet:

• there exists a Γx-equivariant injective ∗-homomorphism

ρ : l∞(Γx)→ l∞(Γ)

defined by ρ(f)(tsi) = f(t) for every t ∈ Γx, sii being a system of representativesof the right cosets Γx\Γ.

• there exists a Γx-equivariant u.c.p. map

ψ : l∞ → C(∂FΓ),

by universal property of ∂FΓ, and the fact that the spectrum of l∞(Γ) is βΓ. (forany compact Γ-space, there exists a Γ-map ∂FΓ→ P (X). take the dual of this mapfor X = βΓ).

• The composition φ = evx ψ ρ defines a Γx-invariant state on l∞(Γx), which con-cludes the proof.

15

Chapter 1. C∗-simplicity

Second bullet:

This needs a lemma:

Lemma 1.5.2. Let X be a Γ-boundary. For every non empty subset of X, every ε > 0,there exists a finite subset F ⊂ Γ\eΓ such that

mint∈F

µ(tU c) < ε ∀µ ∈ P (X).

Proof. Let x ∈ U . By strong proximality, there exists tµ 6= eΓ such that

δx(U)− µ(tµU) = µ(tµUc) < ε,

and by continuity of the action

Vµ = ν ∈ P (X) | ν(tµUc) < ε

is a neighboorhood of µ. By compactness of P (X) in the weak-∗ topology, we can extracta finite cover such that

P (X) = ∪i=1,mVµi .

Then F = tµ1 , ..., tµm fills the requirements of the lemma.

Suppose the action is not topologically free and let s 6= eΓ such that the interior U of Xs

is not empty. Let F the finite subset given by the lemma for U and ε = 13. Suppose

λΓ < λΓ/Γx .

We will show this is absurd by looking at the coefficient cγ = 〈λΓ(γ)δe, δe〉, which is 0unless γ = eΓ.

On the finite subset K = tst−1t∈F , approximate cγ up to ε by a convex combination∑j=1,n

αj〈λΓ/Γx(γ)ξj, ξj〉

of coefficients of the quasi regular representation. Set

µj =∑y∈Γ.x

|ξj(y)|2δy ∈ P (X) and µ =∑

αjµj,

where we identify Γ.x with Γ/Γx. A FINIR

Questions:

• Can we get a more direct proof for the last implication? (without representationtheory)

• It is not known in general wether the action of Γ on ∂FΓ is amenable. If X is a Γ-space such that one of the stabilizer is not amenble, the action cannot be amenable.Is it true that, if Γ is exact, this is the only obstruction for the amenability of theaction?

16

1.6. Another proof

1.6 Another proof

The last subsection uses representation theory (induction) which makes one wonder if thiscould be avoided. While the implication

∂FΓ is free ⇒ Γ is C∗-simple

is still good enough if one wants to stay clear of representation theoretic lingo, the otherdirection can be proven in another way.

This proof is taken from a set of notes that Ozawa wrote after giving lectures for the“Annual Meeting of Operator Theory and Operator Algebras” at Tokyo university, 24–26December 2014.

For X a compact Γ-space and H a subgroup of Γ, we denote by:

• Ex : C(X) or Γ → C∗r (Γ) the canonical conditional expectation onto C∗r (Γ) givenby extending the evaluation at x,

• EH : C∗r (Γ)→ C∗r (H) the canonical conditional expectation given by E(λs) = δs∈H ,

• τH the canoncical trace C∗r (H)→ C.

The first thing one can show is the following.

Proposition 1.6.1. Let X be a Γ-boundary, then

C(X)or Γ

is simple.

Proof. It is enough to show that any quotient map

π : C(X)or Γ→ B

is injective. By C∗-simplicity, π restricts to an isomorphism on C∗r (Γ) so that the canoncialtrace τ is well defined on π(C∗r (Γ). Seeing C as the sub-C∗-algebra of constant functionsin C(∂FΓ), we can extend τ to B.

C(X)or Γ B

C∗r (Γ) π(C∗r (Γ)) C ⊆ C(∂FΓ)

π

φ

∼= τ

Now φ π restricts to a Γ-u.c.p. map C(X)→ C(∂FΓ) which can only be the inclusion.This ensures that

C(X) ⊆ Dom(φ π).

As φ extends τ , φ π is the canonical conditional expectation C(X)or Γ→ C(X) whichis faithful. In particular, π is faithful, and is injective.

17

Chapter 1. C∗-simplicity

Applying this to X = ∂FΓ, we get that C(∂FΓ)orΓ is simple. In that case, every stabilizer

Γx = s ∈ Γ | sx = x ∀x ∈ ∂FΓ

is amenable. Moreover, the strong stabilizer

Γ0x = s ∈ Γ | ∃U neighborhood of x s.t. sU = idU

is a normal subgroup of Γx.(In particular, is is amenable.) In that case, we will apply thefollowing proposition.

Proposition 1.6.2. Let X be a minimal compact Γ-space. If

C(X)or Γ

is simple and there exists x ∈ X such that Γ0x is amenable, then X is topologically free.

Proof. By minimality, topological freeness is equivalent to Γ0x = 1 for some x.

Indeed, if Γ0x = 1 for some x, every non trivial group element cannot fix any neighborhood

of x hence for every s 6= eΓ, we get a sequence of points that converge to x which are notfixed by s. By minimality,

Xs = y ∈ X | sy 6= yis a non empty dense open set of X for every s 6= eΓ. By Baire category’s theorem,

∩s∈Γ\eXs

is dense in X so that X is topologically free.

Let us show that Γ0x = 1. Define a representation

ρ : C(X)or Γ→ B(l2(Γ/Γ0x))

by ρ(fλs)δγΓ0x

= f(sγ.x)δsγΓ0x. It is clearly covariant on the algebraic crossed-product.

To prove ρ extends to the whole crossed-product, i.e. ‖ρ(a)‖ ≤ ‖a‖C(X)orΓ, it is enoughto show that

〈 ρ(a)δΓ0x, δΓ0

x〉 ≤ ‖a‖C(X)orΓ

because δΓ0x

is cyclic. This follows from the fact that the latter is the compositionτ EΓ0

x Ex of 3 u.c.p maps (so contractive).

Pick up x such that Γ0x is amenable and s ∈ Γ arbitrary that fixes some neighborhood of

x: there exists a neighborhood U of x such that s|U = idU . Let f ∈ C(X) be nonzero andsupported in U . Let us compute

ρ(fλs)δγΓ0x.

• If γ.x ∈ U , then sγ.x = γ.x and

ρ(fλs)δγΓ0x

= f(γ.x)δγΓ0x

= ρ(f)δγΓ0x.

• If γ.x /∈ U , f(γ.x) = 0 = f(sγ.x), so that ρ(fλs)δγΓ0x

= ρ(f)δγΓ0x.

This shows that ρ(f(λs− 1)) = 0. By injectivity, λs = 1 and s = eΓ hence Γ0x = 1 and we

are done.

18

1.7. Thompson’s group V is C∗-simple

•

•

•

I1 I2 I3

J1

J2

J3

Figure 1.2: The graph of

(I1 I2 I3

J2 J1 J3

)

1.7 Thompson’s group V is C∗-simple

In this section, we prove that Thompson’s group V is C∗-simple. Recall that V is definedas the group of piecewise linear bijections of [0, 1) with finitely many points of non dif-ferentiability, all of which are dyadic rational numbers. Such a function f is entirelydetermined by two partitions

[0, 1) =n∐i=1

Ii =n∐i=1

Ji

and a bijection

(I1 ... InJσ(1) ... Jσ(n)

). The intervals Ii and Ji are of the type [a, a + 2−n),

with a dyadic rational in [0, 1). Then f is defined on Ii as the only linear increasingfunction applying Ii to Jσ(i).

In order to prove that V is C∗-simple, we will:

• realize V as a countable group of homeomorphisms of the Cantor set;

• use the following result of Le Boudec and Matte-Bon ([1], thm 3.7):

Theorem 1.7.1. Let X be a Hausdorff locally compact space and Γ be a countablesubgroup of Homeo(X). Suppose that for every non empty open subset U ⊂ X,the rigid stabilizer

ΓU = γ ∈ Γ | γx = x ∀x /∈ U

is non amenable. Then Γ is C∗-simple.

Let G be an ample groupoid with compact base space. We also always suppsose thatgroupoids are second countable, Hausdorff and locally compact. Recall that a bisectionU ⊂ G is a set such that s and r are homoeomorphisms when restricted to U . In particular,any open bisection U induces a partial homeomorphism

αU

s(U) → r(U)

x 7→ r s−1|U (x)

19

Chapter 1. C∗-simplicity

The topological full group JGK is defined as the set of bisections U of G such that s(U) =r(U) = G0. The operations are defined by

e = G0, UV = gg′ | g ∈ U, g′ ∈ V s.t. s(g) = r(g′), U−1 = g−1 | g ∈ U.

Recall that a Cantor space is any compact metrizable totally disconnected space withoutany isolated points. It is a standard fact that they are all homeomorphic. A model for Ωis the countable product AX , where

• A is a finite set, often reffered to as the alphabet ;

• X is a countable set.

Then elements of Ω are infinite words indexed by X. Denote by Ωf the set of finite words

Ωf =∐

finite F⊂X

AF ,

then the topology on Ω is the one generated by the cylinders

Ca = w ∈ Ω | w(x) = a(x) ∀x ∈ F = supp(a).

For finite words a ∈ Ωf , l(a) denotes their length, and if F = N, x ∈ Ω, ax denotes theconcatenation of a and x, i.e. the word obtained by first saying a and then x.

Examples:

1. Let Γ a countable discrete group acting on a Hausdorff compact space X by homeo-morphisms. Then JX o ΓK consists of the bisections of the type

S =∐

Ui × γi

where X =∐n

i=1 Ui =∐n

i=1 γiUi.

2. Let Z act on the Cantor space Ω = 0, 1Z by Bernoulli shift

n(ai)i = (ai+n)i ∀n ∈ Z, a ∈ Ω.

Then JΩ o ZK consists of homeomorphisms φ : Ω → Ω such that there exists acontinuous function n : Ω→ Z such that

φ(x) = n(x).x ∀x ∈ Ω.

3. Let Ω = 0, 1N be another model for the Cantor space. Define T : Ω → Ωcontinuous to be the shift

T (a0, a1, ...) = (a1, a2, ...).

Let G2 be the so-called Cuntz or Renault-Deaconu groupoid defined by

(x,m− n, y) | x, y ∈ Ω,m, n ∈ N s.t. Tmx = T ny.

20

1.7. Thompson’s group V is C∗-simple

Exercise: The reduced C∗-algebra of G2 is isomorphic to the Cuntz algebra

O2 = C∗〈s1, s2 | s1s∗1 + s2s

∗2 = 1, s∗1s1 = s∗2s2 = 1〉.

The open setsUa,b = (ax, l(a)− l(b), bx) | x ∈ Ω

define compact open bisections which cover G2 when a, b run across Ωf .

Then JG2K consists of the bisections of the type

S =n∐i=1

Uai,bi

where Ω =∐

i=1,nCai =∐

i=1,nCbi .

If for a ∈ Ωf , Ia = [a, a+ 2−l(a)) ⊂ [0, 1), thenJG2K → V∐n

i=1 Uai,bi 7→(Ia1 ... IanIb1 ... Ibn

)is an isomorphism of groups.

•

•

•

S = U0,01

∐U10,00

∐U11,1 corresponds to

(I0 I10 I11

I00 I01 I1

)I0 I10 I11

I00

I01

I1

Figure 1.3: The isomorphism JG2K ∼= V

The last example realizes V as a countable subgroup of homeomorphsims of Ω. If U = Cais a cylinder for a ∈ Ωf , then the rigid stabilizer VU is isomorphic to V . But V containsa nonabelian free groups, hence is nonamenable. The above theorem ensures that V isthus C∗-simple.

21

Chapter 1. C∗-simplicity

22

Chapter 2

Weakly and non-weakly banddominated operators

2.1 Approximation of band dominated operators

In the following, X denotes a discrete metric space (e.g. Z) with bounded geometry. Thislast requirement means that, for each positive number r, the cardinality of the r-balls isuniformly bounded, i.e. the number Nr = supx∈X |B(x, r)| is finite. For T ∈ B(l2X),define the matrix coefficients of T by

Txy = 〈δx, T δy〉 ∀x, y ∈ X.

Think of T as a matrix (Txy) indexed by X. The propagation of such a T will then bethe (possibly infinite) number

prop(T ) = infd(x, y) | Txy 6= 0.

If the propagation of T is finite, we will say that T is bounded or has finite propagation.Band dominated operators are the norm limits of bounded operators. They form a C∗-algebra C∗u(X), called the uniform Roe algebra of X.

Questions:

1. If T is band dominated, how can we approximate it by bounded operators?

2. How can we recognize when T is band dominated?

The two next numbers will give partial answers to these two questions. Or at least try toexplain why they are not trivial.

2.2 Approximation by bounded operators

For T ∈ B(l2X) band dominated, define T (n) to be the operator with matrix coefficients

T (n)xy =

Txy if d(x, y) ≤ n0 otherwise.

We hope that T n converges to T in norm as n goes to ∞.

23

Chapter 2. Weakly and non-weakly band dominated operators

As an example, take X = Z with its canonical metric (given by the absolute value).Each f ∈ C(S1) gives rise to a multiplication operator Mf ∈ B(L2(S1)), and by Fouriertransform to a convolution operator Tf ∈ B(l2Z). It is the operator of norm ‖f‖∞ with

matrix coefficients (Tf )xy proportional to f(x− y).

In particular, if f =∑N

n=−N λnzn is a trigonometric polynomial, then Tf is bounded as

f(n) = 0 for |n| > N . This ensures that every Tf is band dominated, as every continuousfunctions is a uniform limit of trigonometric polynomials. For such operators, our naiveguess

“ T(n)f →‖‖ Tf ”

is equivalent to

“N∑

n=−N

f(n)zn →‖‖∞ f ”

which is false. It is even worse: one can have ‖Tf‖ = 1 while ‖T (n)f ‖ goes to ∞ (Baire

category argument, see [17] p. 167) and this implies (by uniform boundedness theorem)

that (T(n)f )n does not even converges to Tf in the strong operator topology.

2.3 Weakly band dominated operators

Definition 2.3.1. An operator T ∈ B(l2X) has (r, ε)-propagation if for every subsetsA,B ⊂ X such that d(A,B) > r,

‖χATχB‖ < ε.

T is weakly band dominated if, for every ε > 0, there is r > 0 such that T has (r, ε)-propagation.

Note: Bounded implies weakly band dominated, therefore, weakly band dominated beinga closed condition, band dominated implies weakly band dominated, as the intuition sug-gests.

Question: Does weakly bounded implies bounded?

This was claimed without proof for spaces with finite asymptotic dimension by J. Roe ca’97, and actually proved

• by Rabinovich-Roch-Silbermann in ’00 for X = Zn [11];

• by Spakula-Tikuisis in ’16 for finite asymptotic dimension (and a bit more, finitedecomposition complexity spaces for the curious reader) [15];

• by Spakula-Zhang in ’18 for spaces with property A [16].

We have no counterexamples to this date (25 jan. 2019).

Theorem 2.3.2 (Folklore). The following are equivalent:

1. T is weakly band dominated;

2. for every ε > 0, there exists δ > 0 such that if f ∈ l∞(X)1 and Lip(f) ≤ δ then‖[T, f ]‖ < ε.

24

2.4. Characterizing membership in the Roe algebra

Proof. Let us start with the reverse implication. Say d(A,B) > r, then there existsf ∈ l∞(X)1 satisfying 0 ≤ f ≤ 1, f|A = 1, f|B = 0 and Lip(f) ≤ 1

r. Then fχA = χA and

fχB = 0 so that

χATχB = χA[f, T ]χB

and ‖χATχB‖ ≤ 1r.

Remark: the function

f(x) = max0, 1− d(x,A)

r

does the job. The Lipschitz constant is smaller than 1r

because of the easy inequality

|d(x,A)− d(y, A)| ≤ d(x, y) ∀x, y ∈ X,A ⊂ X.

2.4 Characterizing membership in the Roe algebra

The main goal of this section is to prove the following result, after the work of Spakulaand Tikuisis.

Theorem 2.4.1. Consider the following properties of an operator b ∈ B(l2X).

1. lim ‖[b, fn]‖ = 0 for every very lipschitz sequence (fn) ⊂ Cb(X);

2. b is quasi local;

3. [b, g] ∈ K(l2X) for every Higson function g ∈ Ch(X);

4. b ∈ C∗u(X).

Then (4) =⇒ (1) ⇐⇒ (2) ⇐⇒ (3). Moreover if X has finite asymptotic dimension,then (4) is equivalent to all of these.

Some remarks are in order.

• These results grew out of a question of John Roe, who asked about the implication(2) =⇒ (4) when X has finite asymptotic dimension (FAD).

• The theorem in [15] is better: (2) =⇒ (4) when X has straight finite decompositioncomplexity (FDC), which is much weaker than FAD. For instance, Z o Z has FDCbut not FAD, while FAD always implies FDC.

• There is a follow up paper which shows (2) =⇒ (4) when X has property A, aneven weaker condition. This last result will be treated in a following number.

Let us understand the conditions better.

25

Chapter 2. Weakly and non-weakly band dominated operators

Very Lipschitz condition

Recall that a function f is Lipschitz if its Lipschitz modulus

Lip(f) = supx 6=y

|f(x)− f(y)|d(x, y)

is finite. More precisely, a function f is L-Lipschitz if Lip(f) ≤ L ⇐⇒ |f(x)− f(y)| ≤Ld(x, y), ∀x 6= y.

A sequence (fn) ⊂ l∞(X) is very Lipschitz if

• the sequence is uniformly bounded: ∃C > 0 such that ‖fn‖ ≤ C;

• limLip(fn) = 0.

With this notation, the condition (1) is equivalent to

∀ε > 0,∃L > 0 s.t. if ‖f‖ ≤ 1 and Lip(f) ≤ L then ‖[b, f ]‖ < ε.

Indeed, one direction is obvious, and suppose there exists ε > 0 such that for every L > 0there is a f ∈ l∞(X) with ‖f‖ ≤ 1, Lip(f) ≤ L and ‖[b, f ]‖ ≥ ε. Take L = 1

nto get a

very Lipschitz sequence (fn) with ‖[b, f ]‖ ≥ ε > 0, which contradicts (1).

Quasi-locality

Recall that b ∈ B(l2X) is quasi-local iff ∀ε > 0, b has finite ε-propagation, iff ∀ε > 0,∃r >0 such that ∀f, g ∈ l∞(X), if ‖f‖, ‖g‖ ≤ 1 and d(supp(f), supp(g)) ≥ r then ‖fbg‖ < ε.

Let us introduce the space

CL,ε = a ∈ B(l2X) : ‖[a, f ]‖ < ε ∀f ∈ l∞(X)1 s.t. Lip(f) ≤ L.

What was said above reduces to the fact that the algebra of quasi-local operators is exactly⋂ε>0

⋃L>0

CL,ε.

Higson functions

A function g ∈ l∞(X) is said to be a Higson function, algebra denoted by Ch(X) iff∀ε > 0,∀r > 0, there exists a finite subset F ⊂ X such that if x, y /∈ F and d(x, y) ≤ r,then |g(x)− g(y)| ≤ ε.

2.4.1 Roe’s question on conditions (2) and (4)

(4) =⇒ (2) is not difficult. In short, quasi-locality is a closed condition, which is obvi-ously satisfied by finite propagation bounded operators.

Closed condition If ∀δ > 0, there is a quasi-local operator b′ such that ‖b− b′‖ < δ, thenb is quasi-local.

26

2.4. Characterizing membership in the Roe algebra

Finite propagation operators are quasi-local If ξ ∈ l2(X) is finitely supported and prop(b) ≤r, then supp(bξ) ⊂ Nr(supp(ξ)), and so if d(supp(f), supp(g)) > r, then gbf = 0.

(4) =⇒ (1) is again not too hard.Condition (1) is closed and is satisfied by finite propagation operators. This follows fromelementary estimates and a calculation of the kernel of the commutator.

Lemma 2.4.2. If b ∈ B(l2X) such that |b(x, y)| ≤ C and prop(b) ≤ r, then ‖b‖ ≤ CNr

Lemma 2.4.3. Let b as above and f ∈ l∞(X). The kernel of [b, f ] is

[b, f ](x, y) = b(x, y)(f(x)− f(y))).

Now (4) =⇒ (1) follows: if prop(b) ≤ r, then

prop([b, f ]) ≤ r and |[b, f ](x, y)| ≤ CLip(f)r

so that also ‖[b, f ]‖ ≤ CrNrLip(f). As for the lemmas, the first point reduces to:

|bξ(x)| ≤∑

y∈Br(x)

|b(x, y)| |ξ(y)|

≤ CN12r ‖ξ|Br(x)‖2 by CBC.

=⇒ ‖bξ‖22 =

∑x

|bξ(x)|2

≤∑x

C2Nr‖ξ|Br(x)‖22

≤∑x

∑y∈Br(y)

C2Nr|ξ(y)|2

≤ C2N2r ‖ξ‖2

2

The second point is a direct calculation.

(1) =⇒ (2)

The key point is the following.

Lemma 2.4.4. If A,B ⊂ X such that d(A,B) ≥ r, then there exists a function φ : X →[0, 1] such that

• φ = 1 on A,

• φ = 0 on B,

• Lip(φ) ≤ 1r.

Proof. Let us show that it gives the claimed implication. Let ε > 0, condition (1) givesa constant L. Put r > L−1. If then f, g ∈ l∞(X)1 such that d(supp(f), supp(g)) ≥ r wehave fφ = f and gφ = 0 so that

‖fbg‖ = ‖f [φ, b]g‖ ≤ ‖[φ, b]‖ < ε.

As for the lemma, just take

φ(x) = max0, 1− d(x,A)

r.

27

Chapter 2. Weakly and non-weakly band dominated operators

(2) =⇒ (1)

The key here idea is: if f has a small Lipschitz constant, then it varies slowly so that itslevel sets are well separated.

Let f ∈ l∞(X) such that 0 ≤ f ≤ 1 and Lip(f) ≤ L, and put

Ai =x |i− 1

N< f(x) ≤ i

N i = 2, N

A1 =x |0 ≤ f(x) ≤ 1

N

Then f ∼∑N

i=1iNAi := g (actually ‖f − g‖ ≤ 1

N) and also

d(Ai, Aj) ≥1

NLif |i− j| ≥ 2.

Also the Ai’s are disjoint and cover X. we will now estimate ‖[b, g]‖. Let ε > 0,

‖[g, b]‖ = ‖[∑i

i

NAi, b]‖

= ‖

(∑i

i

NAi

)b

(∑i

Ai

)−

(∑i

Ai

)b

(∑i

i

NAi

)‖

= ‖∑i,j

(i

N− j

N)AibAj‖

≤ ‖∑|i−j|=1

1

NAibAj‖+ ‖

∑|i−j|≥2

(i

N− j

N)AibAj‖

Let us label the summands of this last line by I and II. By quasi-locality of b, we get ar = r(ε) > 0, then for any choice of N , put L = L(N, ε) such that L < (rN)−1. Anysuch f with Lip(f) ≤ L satisfies d(Ai, Aj) > r so that ‖AifAj‖ < ε for each term in thesecond summand, so that

(II) ≤ N2ε.

For (I), the pairs (i, j) can be split up into 4 classes: (i odd, j = i+ 1), (i even, j = i+ 1)and the two symmetric cases. For each of these families, the sum is a block sum withorthogonal domain and range, hence the norm of the sum is less than the sup of the normof the terms, so that

(I) ≤ 4

N.

Let us wrap all of this up: if ε is given, choose N such that 4N< ε, choose L = L(N, ε

N2 ).This gives:

‖[b, g]‖ ≤ (I) + (II)

≤ 4

N+

ε

N2N2

≤ 2ε.

28

2.5. Heart of the paper

2.5 Heart of the paper

Let us turn the attention to the most important result of the paper:

If X has FAD, then (1) =⇒ (4).

Theorem 2.5.1. Let X be a bounded geometry uniformly discrete metric space. If Xhas finite asymptotic dimension, then

∀ε > 0, ∃L > 0 s.t. a ∈ Commut(L, ε) =⇒ a ∈ C∗u(X)

anda ∈ Commut(L, ε) ⇐⇒ ‖[a, f ]‖ < ε ∀f ∈ l∞(X)1, Lip(f) < L.

Review of asymptotic dimension

Recall that X has asymptotic dimension less than d if for every r > 0, there exists abounded cover X which is (d, r)-separated. The typical example is the group Z with themetric induced by the absolute value, which has asymptotic dimension bounded by 1. Asan exercise, prove that asdim(Zn) ≤ n.

In the context of asdim ≤ 1, conditional expectations into block subalgebras are verynatural. Consider subsets Ui of X which are pairwise disjoint and ui the correpondingmultiplication operators. Define

θ(a) =∑i

uiaui ∀a ∈ B(l2X).

• θ(a) is SOT convergent;

• θ is lower continuous;Both of these follow essentially from

‖θ(a)ξ‖2 =∑i

‖uiauiξ‖2 by orthogonality of the support,

≤ ‖a‖∑i

‖uiξ‖2

≤ ‖a‖ ‖ξ‖2

Take the directed systems of all the sums over finite subsets of I, in which case thesum is finite.

• θ is a conditional expectation. (Meaning it is CP, θ(xay) = xθ(a)y when x, y areblock diagonals wrt

⊕i l

2Ui, and θ(a) is block diagonal.)

Write u =∑

i ui.

Fact: if prop(a) ≤ r and U is 2r-separated, then uau = θ(a).

Proof. If ξ ∈ l2X, supp(uiξ) ⊂ Ui, so that supp(auiξ) ⊂ Nr(Ui) which is disjoint from Uj,j 6= i. Hence the cross terms ujauiξ vanish. We get for finite sums∑

i,j∈F

uiaujξ =∑i∈F

uiauiξ,

and the result follows by continuity.

29

Chapter 2. Weakly and non-weakly band dominated operators

Consequence: If b ∈ C∗u(X), for every ε > 0, there exists r > 0 such that if U isr-separated, then

‖ubu− θ(b)‖ < ε.

This renders the next proposition natural.

Proposition 2.5.2. (Cor 4.3) If a ∈ Commut(L, ε), with the notations above, if U is2L

-separated, then‖uau− θ(a)‖ < ε.

Remark: if the theorem is true, then the above discussion shows that the result abovemust be true.

Proof. (of the theorem, assuming the above proposition) If asdim(X) ≤ 1, fix a big r > 0:we get a bounded cover Y which is (1, r)-separated, meaning

Y = U ∪ Vwith U and V r-separated families. Then, if a ∈ B(l2X),

a = uau+ uav + vau+ vav.

we want to show that if a ∈ Commut(L, ε) and r > 4L−1, then each term on the right isnear a finite propagation operator.

Claim: this is true for uau and vav.

This follows from the proposition: U is r > 4L−1 > 2L−1-separated so that

‖uau− θ(a)‖ ≤ ε

and θ(a) is block diagonal w.r.t. a bounded family, it is thus of finite propagation (lessthan supU diam(U)).

Claim: this is true for uav and vau.

Let U ′ = NL−1(U), same for V ′. Both are at least 2L−1-separated. We thus obtainf =

∑fi with fi [0, 1]-valued, with value 1 on Ui, 0 on NL−1(Ui)

c and Lip(fi) ≤ L.Similarly for V , we get g =

∑j gj. Then uf = u and vg = v. Put

Wij = NL−1(Ui) ∩NL−1(Vj) ,W =∐i,j

Wij

which is at least 2L−1-separated. Similarly, build w and wij. we then calculate:

uav = ufagv

= ugafv + u[f, a]gv + u[a, g]fv

So ‖uav − ugafv‖ ≤ 2ε

but now, ugafv ∈ Commut(L, ε), so that the proposition applies using the Wij whichare 2L−1-separated:

‖wugafvw − θw(ugafv)‖ ≤ ε.

But wugafvw = ugafv since wugug and fvw = fv (supp(fv) ⊂ W and supp(ug) ⊂ W ).And θw(ugafv) is block diagonal with finite propagation.

It remains to prove the proposition.

30

2.5. Heart of the paper

Block diagonal symmetries

Lemma 2.5.3. If a ∈ CL,ε and U is a 2L

-separated family of X then

‖[uau, u]‖ < ε

where u =∑ui is our usual notation for the characteristic function of ∪Ui, and u is a

block diagonal symmetry, i.e. an operator of the type∑εiui, εi ∈ −1, 1.

Proof. Extend each ui to a [0, 1]-valued L-Lipschitz function, which is 1 on Ui and zerooutside of NL−1(Ui). Then the fi have disjoint support so that

f =∑i

εifi

satisfies Lip(f) ≤ L, ‖f‖ ≤ 1 and fu = u. But

[uau, u] = u[a, f ]u

which has norm smaller than ε.

The block diagonal symmetries form a topological group (with the SOT topology), iso-morphic to

∏U Z2 endowed with the product topology. It is thus a totally disconnected

compact group, and has a unique Haar probability measure du.

Lemma 2.5.4. Let Z ⊂ X and b ∈ B(l2Z). If

‖[b, u]‖ < ε ∀u block diagonal symmetry

then ‖b−E(b)‖ < ε, where E : B(l2Z)→⊕

l∞(Ui) is the canonical expectation onto theblock diagonal. Furthermore,

E(b) =

∫G

ubu du.

Remark: the example of the two point space is helpful to understand what is happening.Let say

u =

(ε1 00 ε2

)and b =

(x yz w

)then a simple calculation shows(

x 00 w

)=

1

4

∑u∈G

ubu

=1

4

((x yz w

)+

(x −y−z w

)+

(x yz w

)+

(x −y−z w

)).

The estimate is easy:

‖E(b)− b‖ ≤ 1

4

∑‖bu− ub‖ < ε.

Proof. First check that on the group G, the ∗-SOT, SOT and pointwise convergence co-incide. Since the norms are all smaller than 1, we can consider finitely supported vectors,or even basis vectors.

31

Chapter 2. Weakly and non-weakly band dominated operators

Next the integral is understood in the weak sense, meaning that the assertion is

〈E(b)ξ, η〉 =

∫G

〈 ubu ξ, η〉 du ∀ξ, η ∈ l2X.

Ignoring existence, check the following matrix coefficients

〈E(b)δi, δj〉 =

bii if i = j0 else.

and ∫G

〈ubu δi, δj〉 du =

∫G

〈bu δi, u δj〉du

= (

∫G

εiεj du) bij.

But ∫G

εiεj du = P(εi = εj)− P(εi 6= εj)

which is 12− 1

2= 0 if i 6= j, 1 otherwise.

Finally let’s put all the lemmas together to get the proposition.

Proof. Let U be a 2L

-separated family and a ∈ CL,ε.The first lemma gives

‖[uau , u]‖ < ε ∀u ∈ G,

and now uau ∈ B(l2Z) for Z = ∪Ui, so by the second lemma,

‖uau− E(uau)‖ < ε.

The canonical expectation E(uau) is θU(a), and this concludes the proof.

2.6 Property (A)

The last part of the section is devoted to prove the assertion (quasi-locality implies loc-ality) when X has property (A).

Property (A) and its friends

Motivation: let G be a countable discrete group with a bounded geometry left-invariantmetric d. For each A ⊂ X and every r > 0, define the r-corona of A to be the set

∂rA = x ∈ X | 0 < d(x,A) ≤ R.

Here is a possible definition of amenability.

Definition 2.6.1. The group G is amenable if for all r, ε > 0, there exists a finite subsetA ⊂ X satisfying

|∂rA| < ε|A|.

32

2.6. Property (A)

Remark: we don’t suppose the group to be finitely generated. For instance G =⊕

Z Zwith the metric l(n) =

∑i i|ni| is not finitely generated, yet is of bounded geometry and

amenable. If G is finitely generated, one does not need to quantify over r in the definitionand can use ∂A instead.

This definition of amenability makes perfect sense for any bounded geometry metric space.However, it is a bit silly, since for any bounded geometry space X, the space X ∪ N isamenable. Indeed, given r > 0, take Ar = [r, r + N ] ⊂ N. Then |∂rA|

|A| = 2rN+1

is very

small for N large. This definition of amenabilty is thus local (“ something nice happenssomewhere”) when we actually want to say something about the global structure of X.

Definition 2.6.2. The metric space X is uniformly locally amenable, abreviated (ULA)µafter on, if for all r, ε > 0, there exists s > 0 such that for all probability measureµ ∈ Prob(X), there is a finite subset A ⊂ X satisfying

diam(A) ≤ s and µ(∂rA) < εµ(A).

Remarks:

• The strict inequality is important, otherwise take A = ∅.

• The condition would be vacuous without the condition diam(A) ≤ s, with s uniformon all probability measures. Otherwise just take the uniform probability on A forall A: the measure of the r-corona is 0.

• (ULA)µ is equivalent to property (A), see [2].

• If G is a group, then if G is amenable, G is (ULA)µ. The proof is left as an exercisefor the reader.

More definitions.

Definition 2.6.3 ([4]). The metric space X is exact if for all r, ε > 0, there exists s > 0and a partition of unity φii on X such that

• if d(x, y) < r, then ∑i∈I

|φi(x)− φi(y)| < ε,

• diam(supp(φi)) ≤ s for every i ∈ I.

Definition 2.6.4 ([3]). The metric space X has the metric sparsification property, abre-viated MSP after on, if for all r, ε > 0, there exists s > 0 such that for all µ ∈ Prob(X),there exists Ω ⊂ X such that

• µ(Ω) ≥ 1− ε,

• Ω is a r-disjoint union of s-bounded sets.

Theorem 2.6.5 (by everyone above). Exact =⇒ (1) (ULA)µ =⇒ (2) MSP =⇒ (3)

Exact.

The implication (3) is harder, see Sako [14]. The proof is C∗-algebraic: can we find adirect proof?

33

Chapter 2. Weakly and non-weakly band dominated operators

Proof. (1) Given r, ε > 0, let µ ∈ Prob(X), and φi be as in the definition with∑i∈I

|φi(x)− φi(y)| < ε

Nr

.

Hence for each fixed x, ∑y:d(x,y)≤r

∑i∈I

|φi(x)− φi(y)| < ε = ε∑i

φi(x).

As µ is a probability measure,∑x

µ(x)∑

y:d(x,y)≤r

∑i∈I

|φi(x)− φi(y)| < ε∑x

µ(x)∑i

φi(x).

hence there exists an index i0 such that∑x

µ(x)∑

y:d(x,y)≤r

|φ(x)− φ(y)| < ε∑x

µ(x)φ(x).

with φ = φi0 . now write φ =∑aiχFi

where ai > 0 and Fi+1 ⊂ Fi. All the Fi’s are insupp(φ) so their diameter is bounded above by s.∑

x

µ(x)∑

y:d(x,y)≤r

|∑k

ak(χFk(x)− χFk

(y))| < ε∑x

µ(x)∑k

akχFk(x)

∑x

µ(x)∑

y:d(x,y)≤r

∑k

ak|χFk(x)− χFk

(y)| < ε∑x

µ(x)∑k

akχFk(x).

Hence ∑x

µ(x)∑

y:d(x,y)≤r

|χFk(x)− χFk

(y)| < ε∑x

µ(x)χFk(x) = εµ(Fk)

for some k = k0, and for x ∈ ∂rFk0 ,∑y:d(x,y)≤r

|χFk(x)− χFk

(y)| ≤ 1 ≤∑

x∈∂rFk0

µ(x) = µ(∂rFk0).

Set A = Fk0 , thenµ(∂rFk0) < εµ(A).

34

2.6. Property (A)

Quasi-locality and property (A)

The main goal of this section is to provide a proof of (1) =⇒ (4) in the case where Xhas property (A). Let us fix some notations.

For (X, d) a metric space, a partition of unity will be given by a pair (φ,U) where U is acover of X and φ is a map

φ : X → l2(U)1,+ ,

such that x 7→ φU(x) is supported in U for every U ∈ U . (The notation l2(U)1,+ meanspositive elements of norm 1.) If U = Uii∈I , we will identify l2(U) with l2(I).

The characterization of property (A) which we use is the following, obtained by Dadarlatand Guentner in [4].

Theorem 2.6.6. A metric space X is called exact if, for every r, ε > 0, there exists apartition of unity φ : X → l2(U) such that U is uniformly bounded with finite multiplicityand

d(x, y) ≤ r =⇒ ‖φ(x)− φ(y)‖2 ≤ ε.

If X is discrete and of bounded geometry, exactness and property (A) are equivalent.

We will also need to know that property (A) implies the metric sparsification property,which was proven in the last section.

The key idea of the proof relies on an approximation property of quasi-local operators:their norm can be approximated by finitely supported vectors. This means that if b ∈⋂ε

⋃LCL,ε,

‖b‖ = sup‖v‖=1 , diam(supp(v))<∞

‖bv‖.

This relies on the following lemma.

Lemma 2.6.7 ([16], lemma 5.2). For every M,L, ε, there exists s > 0 such that, for everyb ∈ CL,ε with ‖b‖ ≤M , there exists v ∈ l2(X) satisfying ‖v‖ = 1, diam(supp(v)) < s and

‖bv‖ ≥ ‖b‖ − ε.

Proof. (of the result, using the lemma) Let X discrete with bounded geometry and prop-erty (A), and say b ∈ B(l2X) is quasi-local and fix ε > 0. Then there is L > 0 suchthat b ∈ CL,ε and, by the lemma, a s > 0 such that ‖T‖ can be approximated up to ε bys-supported vectors for every T ∈ C2ε,L with ‖T‖ ≤M .

Choose a partition of unity φ with uniformly bounded support and

d(x, y) ≤ s+1

L=⇒ ‖φ(x)− φ(y)‖ ≤ ε.

Let us show that the norm of

a = b−∑i

φibφi =∑i

φi[φi, b]

is small enough.

35

Chapter 2. Weakly and non-weakly band dominated operators

The following computation shows that a ∈ C2ε,L:

‖[a, f ]‖ = ‖[∑i

φi[φi, b], f ]‖

≤ ‖∑i

φi[b, f ]φi‖+ ‖[b, f ]‖

≤ 2ε

where we used ‖∑

i φi[b, f ]φi − φ[b, f ]φ‖ < ε. This follows from the fact that, if ej arepositive contractions with 2

L-separated support, and T ∈ Cε,L, then ‖eTe−

∑i eiTei‖ < ε.

This is not a trivial statement, and was proven in the last section (Cor 5.3 of [16]).

Of course, ‖a‖ ≤ 2M , so we can apply the statement of the first paragraph to a: thereexists a unit vector v ∈ l2X with support F satisfying diam(F ) < s and ‖av‖ ≥ ‖a‖ − ε,and

|∑i

φi(x)(φi(x)− φi(y))bxy| ≤M(∑i

φ2i (x))

12 (∑i

|φi(x)− φi(y)|2)12

≤M‖φ(x)− φ(y)‖2

so that if x ∈ NL−1(F ), ‖φ(x)− φ(y)‖2 ≤ ε, and

|av|x = |∑i,y∈F

φi(x)(φi(x)− φi(y))bxyvy|

≤∑y∈F

|∑i

φi(x)(φi(x)− φi(y))bxy| |vy|

≤ εM∑y∈F

|vy|

≤ εMN12s ‖v‖.

Now ‖av‖2 =∑

x |av|2x ≤M2N2s ‖v‖2ε2 +

∑x∈NL−1

|av|2x, but a being in C2ε,L,

‖χFaχNL−1 (F )‖ < ε

hence ‖av‖2 ≤ (M2N2s + 1)

12‖v‖ε.

It remains to prove the lemma.

Proof. Let b ∈ Cε,L and M = ‖b‖. Let v ∈ l2(X) be a unit vector such that ‖bv‖ ≤‖b‖ − ε

2M(so that ‖bw‖ ≥ ‖bv‖ − ε). Denote by µ the probablity measure on X defined

byµ(x) = |vx|2.

The MSP implies that there is a subset Ω ⊂ X with µ(Ωc) < ε and Ω is a 4L

-separateddisjoint union

Ω =∐

4L

Ωi

of uniformly bounded subsets, i.e. diam(Ωi) < s for all i. Denote by wi = PΩiv, and

w =∑

iwi. Then the condition above says that ‖v − w‖2 < ε and diam(supp(wi)) < s

36

2.6. Property (A)

so if we could approximate ‖b‖ using one of the wi’s, that would end the proof.

There exists fi ∈ l∞(X)1 such that

• Lip(fi) ≤ L,

• supp(fi) ⊂ NL−1(Ωi),

• fi = 1 on Ωi and 0 outside of NL−1(Ωi).

Then f =∑

i fi and 1 − f are also L-lipschitz functions and fw = w. But bw =[b, f ]w + fbw so

‖bw‖ ≤ ε‖w‖+ ‖fbfw‖

≤ 2ε‖w‖+ ‖∑i

fibfiw‖

In the last line, we used that ‖fbf −∑fibfi‖ ≤ ε:

Now, the same trick fib = [fi, b] + bfi entails that

‖∑i

fibfiw‖2 =∑i

‖fibw‖2

≤ ε∑i

‖wi‖2 +∑i

‖bwi‖2

≤ ε‖w‖2 +∑i

‖bwi‖2

so that

(‖bw‖ − 3ε‖w‖)2 ≤∑i

‖bwi‖2 ≤∑i

‖bwi‖2

‖wi‖2‖wi‖2 ≤ sup

i(‖bwi‖2

‖wi‖2)‖w‖2

from which follows that‖bw‖‖w‖

≤ supi

‖bwi‖‖wi‖

+ 3ε.

and

supi

‖bwi‖‖wi‖

+ 3ε ≥ ‖bv‖ − ε‖v‖‖w‖

≥ ‖bv‖ − ε ≥ ‖b‖ − 2ε

so that there exists i0 such that‖bwi0

‖‖wi0

‖ ≥ 6ε, and diam(supp(wi0)) < s.

37

Chapter 2. Weakly and non-weakly band dominated operators

38

Chapter 3

Noncommutative geometry

3.1 Basic objects and constructions

Mainly, I’m interested in ∗-algebras A (and their completions) which are k-algebrasequipped with an involution ∗. Usually, k = C is the field of complex numbers. Avery famous example of ∗-algebra is the algebra of the quantum harmonic oscillator,

H = k〈x, y〉/(xy − yx = 1).

When k = C, one often represent A as a sub-∗-algebra of the bounded operators on aHil-bert space L(H),and complete w.r.t. to the norm. Note that not all complex ∗-algebrasadmit such a representation.

For instance, forH, one easily get that

[x, P (y)] = P ′(y) ∀P ∈ C[t]

Then if || || is a multiplicative norm onH, it satisfies

2||x|| ||y|| ≥ n ∀n > 0.

Basic construction:

• separation-completion: in our sense, a norm can be degenerate. Being multiplicat-ive, the annhiliator of any norm is a closed ideal in A, so that there is an induced(classical/ nondegenerate) norm on the quotient algebra. The separation-completionis defined to be the completion of the quotient w.r.t. the induced norm. Let us saythat if α is such a norm, wedenote by Aα the associated separation-completion. Anyinequality

α(x) ≤ β(x) ∀x ∈ A

induces an inclusion of annilihator Nβ ⊂ Nα, and gives a canonical quotient map

Aβ → Aα.

The basic class of examples comes from completion of the complex group ring C[Γ].For any family of unitary representations F , one can define the ∗-norm

||x||F = sup||π(x)|| : π ∈ F

39

Chapter 3. Noncommutative geometry

on C[Γ]. The separation-completion is a C∗-algebra denoted C∗F(Γ). For instance, ifF consists of all unitary representations of Γ, then one gets the maximal C∗-algebraC∗max(Γ), while if the family is reduced to the left regular representation λΓ, one getsthe reduced C∗-algebra C∗r (Γ). By inclusion, one gets the canonical quotient map

λΓ : C∗max(Γ)→ C∗r (Γ).

Crossed-product: the basic ingredients are a ∗-algebra H endowed with a coassociativecoproduct

∆ : H → H ⊗H,

and a C∗-algebra A on which H acts via a ∗-homomorphism

α : A→ A⊗H

such that (1 ⊗ ∆)α = (α ⊗ 1)α. The crossed-product is a twisted version of the tensorproduct.

(a⊗ x)(a′ ⊗ y) := (a⊗ 1M(H))α(a′)(1M(A) ⊗ xy)

3.2 Quantum groups

A C∗-bialgebra is a pair (H,∆) where H is a C∗-algebra and

∆ : H →M(H ⊗min H +H ⊗min H,H ⊗min H)

is a non-degenerate ∗-homomorphism such that (1⊗∆)∆ = (∆⊗ 1)∆.

A H-algebra is a pair (A,α) where A is a C∗-algebra and

α : A→M(A⊗min H,A⊗min H)

such that (α⊗ 1)α = (1⊗∆)α). Its principal map is

Ψ :

A⊗alg A → M(A⊗min H)

x⊗ y 7→ (x⊗ 1M(H))α(y)

Let (H,∆) be a C∗-bialgebra and (A,α) a H-algebra, with principal map

Ψ : A⊗ A→M(A⊗min H).

• free if the range of Ψ is strictly dense in M(A⊗min H)

• proper if the range of Ψ is contained in A⊗min H

• principal if Ψ(A⊗alg A) is a norm dense subset of A⊗min H

principal = free and proper

40

3.2. Quantum groups

3.2.1 Why SUq(2)?

Apparently, some people are interested in deformation of classical Lie groups such asSUq(2), which is the Hopf algebra generated by 3 generators E,F,K satisfying the rela-tions

R.

I wanted to understand where these relations are coming from, which led me to interestingideas developed by several people, including Yuri Manin. The idea is to define SUq(2) asa special group like object of the automorphism group of some noncommutative space,the quantum plane.

Let k be a field. The free (noncommutative) k-algebra on n generators is denoted byk〈x1, ..., xn〉.

Definition 3.2.1. A quadratic algebra

A = ⊕i≥0Ai

is a N-graded finitely generated algebra such that:

• A0 = k, and A1 generates A,

• the relations on generators are in A1 ⊗ A1.

The quadratic algebra A is said to be a Frobenius algebra of dimension d if moreover

• Ad = k and Ai = 0 for all i > d,

• the multiplication mapm : Ai ⊗ Ad−i → Ad

is a perfect duality.

The main example is the quantum plane

A2q = k〈x, y〉/(xy − qyx)

where q ∈ k×. More generally, the quantum space of dimension n|m is

An|mq = k〈x1, .., xn, η1, ..., ηm〉/(xixj − qxjxi, qηiηj + ηjηi).

This example is suppose to come from physics. In quantum field theories, physicists dealwith two kind of particles, bosons and fermions, and use commuting variables for one type,and anticommuting for the other. One object they appeal to are called supermanifolds,which are manifolds enriched with anticommuting variables. Formally, it means they lookat ringed spaces (X,O) locally isomorphic to (Rn, C∞[η1, ..., ηm]), where C∞[η1, ..., ηm] isthe free sheaf of rings generated by anticommuting variables ηi over the smooth complexvalued functions C∞(Rn).

Remark that a quadratic algebra A is a quotient of k〈x1, ..., xn〉 by elements rα ∈ A1⊗A1,which we will denote as

A = k〈x1, ..., xn〉/(rα)

41

Chapter 3. Noncommutative geometry

orA = 〈A1, RA〉

with RA ⊆ A1 ⊗ A1.

Manin defines the quantum dual of a quadratic algebra as

A! = k〈xi〉/(rβ)

where rβijrijα = 0, i.e. RA! = R⊥A. Then, the quantum endormorphisms between two

quadratic algebra isHom(A,B) = k〈zji 〉/(rβα)

where rβα = rijα rβklz

ki z

lj. If End(A) = Hom(A,A), then End(A) satisfies the universal

property to be intial in the category of k-algebras (B, β) endowed with an algebra homo-morphism β : A→ A⊗B.

If one does that to the quantum plane A2q, one stil doesn’t find quite Mq(2): half of the

relations are missing. Also(A2|0

q )! = A0|2q ?

Exercise.

3.3 TQFT

3.3.1 Motivations

This section is aimed at being an introduction to Topological Field Theories. One ofthe difficulties of this particular topic is that it comes from different areas and can beattacked in different ways. The following is my attempt to make sense out of the largeamount of information available on the subject. In particular, I do not claim exhaustivityor expertise.

The starting point are probably path integral formulations in Physics. In StatisticalMechanics and in Quantum Physics, the values predicted by the theory can often bewritten as expectation of the type

E[exp(−∫V (q(t))) or E[exp

i

hS(q)].]

Physically, one tries to define a positive function on the phase space M (such as anenergy (Hamiltonian) H or an action S, the integral of a Lagrangian). The probabilitydistribution of the system should then be

1

ZMe−βH(ω)Dω or

1

ZMe

ihS(ω)Dω.

The first case is the one known as Gibbs measures, and describes the behaviour of asystem in contact with a thermostat at inverse temperature β. The second case is theso called Feynman integral of quantum mechanics. The reason these formulas are usedis that systems should satisfy some minimization principle. In the classical case, the ob-served trajectories are the minima of the energy function, whereas in the quantum case

42

3.3. TQFT

the observe deviations from the classical trajectories up to the amplitude eiS/h.

That point is exactly where it starts to be complicated. Physicists want to define some-thing propoptional to these exponential, and the measure Dω is supposed to be a referencemeasure with nice invariance properties. In the finite dimensional case, the natural meas-ure would be the Lebesgue measure. But no such thing exists on a general functionalspace, which makes the definition above useless.

It turns out that these integral have very interesting invariance properties, notably intopology. More precisely, the partition function ZM gives topological invariant whenM is a closed manifolds. This fact gave motivation to mathematicians to study moreattentively these functional integrals. While you can try to define integrals analytically(see REFERENCES), there exists an algebraic approach which proposes intuitively todefine the partition function on simple manifolds (possibly with border) and in coherentmanner so that the partition function of a closed manifold can be computed by cutting itin simple pieces, computing the corresponding values, and reassembling these to get thefinal result. Mathematically, this requires:

• describing an algebraic structure on the family of n-dimensional manifolds, andgiving generators;

• setting up the value of the partition function;

• showing that all of this makes sense.

This is accomplished by considering the n-category of bordisms and defining the (algeb-raic) partition function Z to be a nice functor between the latter and the n-category offinite dimensional vector spaces over some fields. In dimension 1 and 2, it will be under-stood that only one value needs to be fixed (respectively on the point space and on thecircle), while this still holds in higher dimension under more hypothesis.

3.3.2 Summary of the talks

We recalled the definitions of a monoidal category, a braided category, and a symmet-ric monoidal category. The two main examples are the category of bordisms Bordd indimension d, and the category of vector spaces over a field k. The first talk focused ontopological quantum fields theories in dimension 1 and 2.

Definition 3.3.1. A TQFT in dimension d is a monoidal symmetric functor

Z : Bordd → V ectk.

The two main results we showed are:

• there is an equivalence of categories

TQFT1∼= V ectk

obtained as Z 7→ Z(pt).

43

Chapter 3. Noncommutative geometry

• there is an equivalence of categories

TQFT2∼= Frobk

obtained as Z 7→ Z(S1).

A nice example in dimension 2: Z(S1) = C[t]/(t2 − 1) is the Frobenius algebra given by

∆(t) = 1⊗ t+ t⊗ 1 ε(1) = 0 ε(t) = 1.

Then the handle element is h = 2t and

Z(Σg) =

2g if g is odd0 if g is even.

The second talk was directed towards extended field theories. First recall some highercategory theory: n-categories, etc... And an extented TFT is a symmetric monoidalfunctor between symmetric monoidal n-categories

Z : Cobn → C.Then the following theorem was proved in [8].

Theorem 3.3.2. The evaluation functor

Z 7→ Z(∗)establishes a bijective correspondance between extended n-dimensional TFT and fullydualizable objects of C.We now give an application of this result to the Jones polynomial. In [21], Witten givesan interpretation of the Jones polynomial, an isotopy invariant of links, as induced froma 3-dimensional TFT. The drawback of this article (for us) is that Witten uses PhysicalTFT’s, i.e. gauge theories. The Jones polynmial is then shown to be the value of thepartition function of a gauge field theory on S3 with gauge group SU(2). I propose torewrite this result in our setting as an exercise.

A link is a disjoint union of embedding of the circle into S3

L = embeddingsk∐i=1

S1 → S3.

we will often make no distinction between the embedding and its image in the 3-sphere,which we will denote by L. The Jones polynomial of a link L is defined as an isotopyinvariant polynomial V : L → Z[t

12 , t−

12 ] satisfying the Skein relations

−t12V+ + (t

12 − t−

12 )V0 + t−

12V− = 0.

To a link L one can associated the 3-manifold ML = S3−L. Consider the extended TFT

Z(n) : Cob3 → Cgiven by Z() = Vn where is the fundamental representation of su(n). By the cobordismtheorem, it is enough to define the TFT on all of Cob3. Then

φ(VL) = Z(2)(ML),

where φ : Z[t12 , t−

12 ]→ C is the evaluation at a root of unity q ∈ C×. This can be proved

by showing that Z(n)(ML) satisfies the skein relation

−qn2 V+ + (q

12 − q−

12 )V0 + q−

n2 V− = 0

44

3.4. Reminder

3.4 Reminder

A locally ringed space is a topological space X together with a sheaf or ring OX over Xsuch that all stalks are local rings, ie have a unique maximal ideal.

For R a ring, X = Spec(R) denotes the topological space obtained as the set of primeideals of R endowed with the Zariski topology, i.e. the topology generated by the closedsubsets

VI = J ideals in R s.t. I ⊂ J.

Equivalently, a basis of open subsets is given by

Df = J ideals in R s.t. f /∈ J

for every f ∈ R. Let Sf be the multiplicative domain given by the powers of f . Thendefine a sheaf of ring over X by

OX(Df ) = S−1f R.

It is called the structural sheaf of Spec(R). Any locally ringed space isomorphic to

(Spec(R),OSpec(R))

with R commutative is called an affine variety.

Note: the functor Spec gives an antiequivalence of categories between the categories ofcommutative rings and the category of affine varieties.

Definition 3.4.1. A scheme is a locally ringed space locally isomorphic to an affinevariety.

45

Chapter 3. Noncommutative geometry

46

Chapter 4

Dynamical Property (T)

The first thing I will try to do is to justify the use of groupoids. My opinion is thatthese objects are not loved as much as they deserve. People who very much like short andconcise definitions enjoy to say that groupoids are small categories in which all morphismsare invertible. This is true, but maybe does not shed light on the reasons people look atsuch objects.

Groupoids can be thought as a generalisation of both groups and spaces. In that effect,a groupoid G is made of two parts, in our case, two spaces, the group-like part G andthe space-like part G0. Usually G is called the space of arrows, and G0 the base space,seen as a subset of G. Any arrow g ∈ G has a starting point x ∈ G0 and an ending pointy ∈ G0. This is encoded by two maps s, r : G⇒ G0 called source and range. Two arrowscan be composed as long as the ending point of the first coincides with the starting pointof the second. The points of the base space act as units, and every arrow as an inversewith respect to this partial multiplication.

g

g−1

•s(g) •r(g)

In our setting, all the spaces will be topological spaces and the maps will be continuous.We will even simplify greatly our life by only looking at second countable, locally compact,etale groupoids with compact base space. From now on, we will only say etale, forgettingabout all other technical assumptions to gain in clarity.

Being etale means that the range map r : G → G0 is a local homeomorphism, i.e. forevery g ∈ G, there exists a neighborhood U of g such that r|U is a homeomorphism. Thisimplies in particular that every fiber Gx = r−1(x) and Gx = s−1(x) are discrete. Whenthe base space G0 has the additional property of being totally disconnected, we will saythat G is ample. Here is a list of examples of etale groupoids.

• A (nice) compact space X defines a trivial groupoid G = G0 = X and source andtarget are the identity; in the opposite direction if the base space is a point, thegroupoid is a group. One can already see how the notion of groupoid generalises

47

Chapter 4. Dynamical Property (T)

both spaces and groups as promised.

• As an intermediate situation between these two cases, consider a discrete group Γacting by homeomorphisms on a compact space X. Define the action groupoid asfollow. Topologically, it is the space G = X × Γ ⇒ G0 = X. The multiplicationencodes the action

(x, g)

(g.x, g−1)

(g.x, g′)

•x • g.x •g′g.x

and this picture gives every element to reconstruct the groupoid.

• If R ⊆ X×X is an equivalence relation, then R as a canonical structure of groupoidwith the base space being the diagonal R0 = (x, x) | x ∈ X and the multiplicationbeing the only one possible

(x, y)(y, z) = (x, z).

• More interesting is the coarse groupoid G(X) associated to a discrete countablemetric space (X, d) with bounded geometry, that is

supx∈X|B(x,R)| <∞ ∀R > 0.

A nice way of thinking about this condition is to imagine yourself looking at thespace with a magnifying glass of prescribed radius, but as great as you wish. Thenyou should not observe more and more points in your sight as you move around. Inother words, the points fitting in the radius of your glass is uniformly bounded.

Now consider the R-diagonals:

∆R = (x, y) | d(x, y) <∞ ⊆ X ×X

and take their closure ∆R in β(X ×X) (βY being the Stone-Cech compactificationof Y ). The coarse groupoid is defined topologically as

G(X) = ∪R>0∆R ⇒ βX,

and is endowed with the structure of an ample groupoid which extend the groupoidX×X ⇒ X associated with the coarsest equivalence relation on X. The topologicalproperty of this groupoid encodes the metric or coarse property of the space. Forinstance, X has property A iff G(X) is amenable, X is coarsely embeddable into aHilbert space iff G(X) has Haagerup’s property, etc.

48

• The last construction is associated to what is often referred as an approximatedgroup, which is the data of N = Γ, Nk where Γ is a discrete group, and theNk’s are a tower of finite index normal subgroups with trivial intersection, i.e.

N1 / N2 / ... s.t. ∩k Nk = eΓ and [Γ : Nk] <∞.

Then the Γk’s are finite groups. Set Γ∞ = Γ for convenience (which is not usuallyfinite!). For any discrete group Λ, there exists a left-invariant proper metric, whichis unique up to coarse equivalence (take any word metric if the group is finitelygenerated). Let us denote by |Λ| the coarse class thus obtained. Then the firstobject of interest in that case is the coarse space XN defined as the coarse disjointunion

XN =∐k

|Γk|.

Here the metric is such that d(|Γi|, |Γj|)→∞ as i+ j goes to ∞, i 6= j.

The second interesting object attached to N is the HLS (after Higson-Lafforgue-Skandalis [5], where it was first defined to build counter-examples to the Baum-Connes conjecture) groupoid. The base space is the Alexandrov compactification ofthe integers

G0N = N,

and GN is a bundle of groups with the fiber of k being Γk. The topology is takento be discrete over the finite base points, and a basis of neighborhood of (∞, γ) isgiven by

Vγ,N = (k, qk(γ)) | k ≥ N N ∈ N,

where qk : Γ→ Γk is the quotient map.

One of the reasons we use groupoids is that they are convenient to build interestingC∗-algebras. To see their relevance, one may start with the question What are operatoralgebraists doing? A possible answer is that part of Noncommutative Geometry andOperator Algebras are devoted to the construction of interesting classes of C∗-algebras.For instance, nuclearity was naturally introduced after Grothendieck’s work, followed bya C∗-algebraic formulation. Arises then the question does there exist nonnuclear C∗-algebras? A now classical result states that, when Γ is a discrete group, the reducedC∗r (Γ) is nuclear iff Γ is amenable. Calling out a nonamenable group, like any nonabelianfree group, produces then a nonnuclear C∗-algebra. This game revealed itself to be veryfruitful: study a property in some field and try to apply it to C∗-algebras to see whatexotic being can be built out of it. The most common fields that have natural C∗-algbrasassociated to them are traditionally group theory, coarse geometry and dynamical systems(there are others like foliations etc, but let me just limit myself to these ones). This canbe summarized in the following diagram.

49

Chapter 4. Dynamical Property (T)

C∗-algebras

Groupoids

CoarseGeometry

G(X)

Topologicaldynamics

Ωo Γ

Group theoryCay(Γ, S)Γ y Ω

Another interesting strategy is to try and translate a property in one of those upper boxesdirectly in terms of groupoids. Then the property can either be used to build C∗-algebras,either give a new definition in the case of other upper boxes. For instance, that is what wetried to do with Rufus Willett in our work on property T. Property T is originally a groupproperty defined in terms of its unitary representations. In [20], Willett and Yu defineda geometric property T for monogenic discrete metric spaces with bounded geometry.Following their work, our first goal was to try and define a property T for (nice enough)topological groupoids so that in the case of groups and coarse groupoids, it reduces tothese notions of property T. It gives then a notion of property T for dynamical systems,by considering property T for the action groupoid X o Γ. The second part of the workis dedicated to go down the last arrow, that is studying implications of property T for Gto its reduced and maximal C∗-algebras, and even more general completions of Cc(G).

Let us first recall what is property T for discrete groups.

If π : Γ→ B(H) is a unitary representation of Γ on a separable Hilbert space, say that πalmost has invariant vectors if for every pair (F, ε) where F is a finite subset of the groupand ε a positive number, there exists a unit vector ξ ∈ H such that

‖s.ξ − ξ‖ < ε ∀s ∈ F.

Definition 4.0.1. A group Γ has property T if every representation that almost hasinvariant vectors admits a nonzero invariant vector.

This definition is not the original one. Indeed property T was defined by Kazhdan inorder to prove that some lattices in some Lie groups were finitely generated. It seemed avery specific property and application, but it turned out that property T gave very niceapplications. Here are some of the most spectacular the author is aware of.

• Margulis supperrigidity theorem (about this, see Monod’s [9] beautiful generaliza-tion, which Erik called the most beautiful paper he ever read);

• existence of expander: for any infinite approximated group (in the sense of the ex-amples above) Γ, the space XN is an expander;

50

• existence of Kazdhan projections which are very wild objects one should only ap-proach with care;

• more generally, property T was for a long time an obstruction to the Baum-Connesconjecture, up until the work of Lafforgue ([7], [6]). It still gives interesting proper-ties for diverse crossed-product constructions as we will see.

One can prove easily that finite groups have T. Indeed, in that case, take the finite subsetto be the whole group and look intensely at the identity

‖s.ξ − ξ‖2 = 2(1−Re〈s.ξ, ξ〉).

If ξ is (Γ, ε)-invariant for ε sufficiently small, then the above identity implies that 1|Γ|∑

s∈Γ s.ξis nonzero because its inner-product with ξ will have real part close to 1. But ξ is invariant.

Now take Γ = Z and look at the left-regular representation, i.e. H = l2Γ and

(s.ξ)(x) = ξ(s−1x).

Then if ξn = 1|Fn|χFn ∈ H is the characteristic function of Fn normalized to be a unit

vector, one can check that

sups∈F‖s.ξn − ξ‖ → 0 as n→∞

so that the regular representation always almost has invariant vectors. But it never hasnonzero invariant ones, so that Z does not have T. This proof actually works for everyinfinite amenable group.

The moral of this story is that if one wants to find infinite groups with property T, onehas to look at nonamenable groups. Maybe F2 or SL(2,Z)? Actually not: they bothsurject to Z which does not have T, and this is an obstruction to having T as is obviousfrom the definition.

Finding infinite groups with property T is actually a hard problem. Here are some ex-amples, without any proofs since these would go out of scope for these notes.

• SL(n,R) and SL(n,Z) if n ≥ 3;

• Sp(n, 1) and its lattices, which gives examples of infinite hyperbolic (in the sense ofGromov) groups having property T;

• Aut(F5) and Out(F5) by a recent result of Nowak and Ozawa [?]. Their proof isinteresting in that they use numerical computations to reach their result using aprevious result of Ozawa [?];

51

Chapter 4. Dynamical Property (T)

• SO(p, q) with p > q ≥ 2 and SO(p, p) with p ≥ 3. More generally, any real Liegroup with real rank at least two, and all their lattices. Also, any simple algebraicgroup over a local field of rank at least two have T.

To define property T for groupoids, we need to choose what kind of representations weare looking at, and to decide what are the invariant vectors.

A representation will be a ∗-homomorphism π : Cc(G)→ B(H). A vector ξ ∈ H is calledinvariant if

f.ξ = Ψ(f).ξ ∀f ∈ Cc(G).

The subspace of invariant vectors is denoted by Hπ and its orthogonal complement, thespace of coinvariants, is denoted by Hπ.

Here Psi... Groups

Let F be a family of representations.

Definition 4.0.2. G has property T if there exists a pair (K, ε) where K ⊆ G is compactand ε > 0 such that, for every π ∈ F , there exists f ∈ CK(G) such that ‖f‖I ≤ 1 and

‖f.ξ −Ψ(f).ξ‖ < ε‖ξ‖ ∀ξ ∈ Hπ.

The first thing we did was to study what were the relationships between groupoid prop-erty T and other property T.

• if G = Γ is a discrete group, Γ has property T iff G has property T (in the groupoidsense);

• if X is a coarsely geodesic metric space, then X has geometric property T iff G(X)has property T;

• in the case of a topological action, X o Γ has property T iff Γ has T w.r.t. thefamily FX of representations π : C[Γ] → B(H) s.t. there exists a representationρ : C(X) → B(H) such that (ρ, π) is covariant. This hypothesis simplifies in thecase where there exists a invariant ergodic probability measure on X; in that caseproperty T for X o Γ and for Γ are equivalent;

• in the case of an approximated group Γ, then GN has property T iff Γ has T. Thismay sound disappointing, but if one refines the result, one gets the nice followingproperty: Γ has property τ w.r.t. N iff GN has T w.r.t. the family of representa-tions that extend to the reduced C∗-algebra of G.

The last part of the work is devoted to the existence of Kazdhan projections. Recall, ifF is a family of representations, C∗F(G) is the C∗-algebra obtained as the completion ofCc(G) w.r.t. the norm

‖a‖F = supπ∈F‖π(a)‖.

52

4.1. Kazdhan projections and failure of K-exactness

A Kazdhan projection p ∈ C∗F(G) is a projection such that its image in any of the rep-resentations in F is the orthogonal projection on the invariant vectors.

Theorem 4.0.3. Let G be compactly generated. Then if G has property T w.r.t. F ,there exists a Kazdhan projection p ∈ C∗F(G).

This gives an obstruction to inner-exactness. Denote by F the closed G-invariant subset

x ∈ G0 | Gx is infinite

and U its complement.

Theorem 4.0.4. Let G be compactly generated and with property T. If one can finda sequence of points (xi)i ⊂ U such that, for every compact subset K ⊂ G, K onlyintersects a finite number of orbits G.xi = r(s−1(xi)), then G is not inner-exact. In factit is not K-inner-exact. in particular, at least one of the groupoids G, G|U or G|U doesnot satisfy the Baum-Connes conjecture.

4.1 Kazdhan projections and failure of K-exactness

For K ⊂ G, CK(G) denotes the continuous functions supported in K.

Theorem 4.1.1. Let G be an etale groupoid whose reduced C∗-algebra contains a nontrivial Kazdhan projection p. Suppose there exists an invariant probability measure onG0 and that there exists an open subset U ⊂ G0 not equal to G0 containing a sequenceof points (xi) such that:

• xi has finite orbit (xi ∈ G0fin);

• for every compact K ⊂ U , the orbits Gxi = r(Gxi) ultimately don’t intersect K;

then C∗r (G) is not K-exact.

Proof. Denote by Mi the finite dimensional C∗-algebra B(l2Gxi) and λi : C∗r (G) → Mi

the corresponding left regular representation. We will show that the sequence

0 C∗r (G)⊗⊕Mi C∗r (G)⊗∏Mi C∗r (G)⊗

∏Mi/⊕Mi 0

q

is not exact in K-theory. We shall call q the last map in this diagram.

Define the following ∗-morphism

φ

C∗r (G) → C∗r (G)⊗ (

∏Mi)

x 7→ x⊗ (λi(x))i

Claim: the image of φ is contained in the kernel of q.

Let x ∈ C∗r (G) and ε > 0. Let K ⊂ G be a compact subset and a ∈ CK(G) suchthat ‖x − a‖r < ε. Let φi be the ∗-homomorphism defined in the same fashion as φonly with the first i components of φ(x) equated to zero. Denote by x the class of x in

53

Chapter 4. Dynamical Property (T)

C∗r (G)⊗∏Mi/⊕Mi. Then φ(x) = φi(x). Also, as the orbits Gxi are ultimately disjoint,

there is a i0 such that λi(a) = 0 and thus φi(a) = 0 for all i > i0. This ensures

‖φ(x)‖ = ‖φi(x)‖ = ‖φi(x)− φi(a)‖ < ε

hence φ(x) = 0.

Let p ∈ C∗r (G) the Kazdhan projection. Then P = φ(p) goes to zero in the right sideof the sequence above. Let us show that its class in K-theory does not come form anelement in K0(C∗r (G)⊗⊕Mi).

The invariant probability measure on G0 induces a trace τ on C∗r (G). Define τi to bethe trace τ ⊗ tr on C∗(G) ⊗Mi, where tr is the normalized trace on Mi. It is easy tosee that τn(P ) = τ(p) > 0. But if z ∈ K0(C∗r (G) ⊗ ⊕Mi), τn(z) is ultimatley zero. Thisimplies that the non triviality of P ensures the non K-exactness of the sequence above inK-theory.

This result gives interesting examples of non K-exact C∗-algebras:

• if X is an expander, the coarse groupoid of X satisfies the hypothesis above, so thatthe uniform Roe algebra C∗u(X) ∼= C∗r (G) is not K-exact; in particular, if Γ containsan expander almost isometrically, its reduced errrrr no?

• if Γ is a residually finite group with property (τ), then any HLS groupoid associatedto an approximating sequence of Γ satisfies the hypothesis above so that C∗r (G) isnot K-exact.

54

Chapter 5

Cartan subalgebras

The goal of this section is... Historical remarks: aVN and Feldman-Moore,...

The first part will detail J. Renault’s work [12] on Cartan pairs.

Recall that an element x ∈ A normalizes a self-adjoint subspace B of A if

xBx∗ ∪ x∗Bx ⊂ B.

The normalizer NA(B) is the set of all the elements of A that normalize B.

Definition 5.0.1. Let A be a C∗-algebra. A sub-C∗-algebra B ⊆ A is called a Cartansubalgebra of A if:

• B is a maximal abelian self-adjoint subalgebra (MASA) of A;

• B contains an approximate unit for A;

• the normaliser of B in A generates A as a C∗-algebra;

• there is a faithful conditional expectation E : A→ B.

The pair (A,B) is referred to as a Cartan pair.

Examples:

• Dn ⊂Mn(C),

• C(X) ⊂ C(X)o Γ,

• l∞(X) ⊂ C∗u(X),

• C0(G0) ⊂ C∗r (G,Σ).

Renault obtained the following result in [12].

Theorem 5.0.2. Any Cartan pair (A,B) is isomorphic to the Cartan pair

(C∗r (G,Σ), C0(G0)),

where G is an etale topologically prinicipal groupoid with base space G0 and Σ is a twistover G.

55

Chapter 5. Cartan subalgebras

This theorem is very uselful. For instance, it implies that a nuclear C∗-algebra with aCartan subalgebra satisfies the universal coefficient theorem of Rosenberg and Schochet[13]. Indeed, the reduced C∗-algebra of an etale groupoid is nuclear iff it is amenable, inwhich case it belongs to the bootstrap class [18].

The first step in the proof of the theorem is to build, for any inclusion of C∗-algebrasA ⊆ B with B unital commutative, an action of NA(B) by partial homeomorphisms onthe spectrum of B. A standard construction then give rise to an etale groupoid GB (thegroupoid of germs of a pseudogroup) of this action. The twist is given by the same kindof construction.

For the second step, one defines a generalized Gelfand transform

5.1 Groupoids of germs

Out of any inclusion of C∗-algebras A ⊆ B with A unital commutative, we construct anaction of the normalizer of A in B by partial homeomorphism on X the spectrum of A,i.e. a homomorphism of semigroup

α : NB(A)→ SHomeo(X).

If n ∈ NB(A) and x ∈ Spec(A), set

〈αn(x), a〉 = 〈x, n∗an〉.

This defines a homeomorphismαn : Un → Un∗ ,

where Un = x ∈ Spec(A), n∗n(x) > 0 such that αnm = αn αm.

Lemma 5.1.1. IfB is abelian and contains an approximate unit, α : NA(B)→ PHomeo(X)is a homorphism of inverse-semigroups.

In our case, given a Cartan pair (A,B), and X = Spec(B), one defines:

• ΣB as the quotient of

(x, n) ∈ X ×NA(B) s.t. n∗n(x) > 0

by the equivalence relation (x, n) ∼ (x, n′) when there exist b, b′ ∈ B such thatnb = n′b′;

• GB as the groupoid of germs of the pseudogroup α(NA(B));

5.2 Generalized Gelfand transform

If (x, n) ∈ X ×NA(B) such that n∗n(x) > 0, and a ∈ A then

E(n∗a)(x)√n∗n(x)

56

5.3. Roe algebras

only depends on the class of (x, n) in ΣB, hence defines a continuous section a of the twistΣB. The map extends to a ∗-homomorphism

Ψ : A→ C∗r (GB,ΣB)

which is always linear injective and respects the Cartan algebras. Moreover, restricted toB, Ψ coincides with the Gelfand transform B → C0(X).

When (A,B) is a Cartan pair, Ψ is an ∗-isomorphism.

5.3 Roe algebras

In the case of uniform Roe algebras, White and Willett have obtained in [19] rigidityresults. The questions are:

• What form can a Cartan subalgebra of C∗u(X) take?

• Can we describe when it is unique up to unitary equivalence?

The answers they have are the following. If X is an infinite countable metric spacewith bounded geometry, any Cartan subalgebra of C∗u(X) is non separable and containsa complete family of orthogonal projections. Interesting examples show that Cartansubalgebras of uniform Roe algebras need not be isomorphic to l∞. Let us recall:

Definition 5.3.1. A sub-C∗-algebra B of A is a Roe Cartan pair if:

• A is unital;

• A contains the C∗-algebra of compact operators on a separable infinite dimensionalHilbert space as an essential ideal;

• B is a co-separable Cartan subalgebra of A abstractly isomorphic to l∞(N). (co-separable means that there is a countable subset of A which generates A togetherwith B)

Theorem 5.3.2. Let (A,B) a Roe Cartan pair. Then there exists a metric space withbounded geometry X such that for any irreducible faithful representation of A on a Hilbertspace H, there exists a unitary u : l2(X) → H that conjugates A with C∗u(X), and Bwith l∞(X).Moreover, if A = C∗u(Y ) for some bounded geometry metric spaxce Y with property A,then X and Y are coarsely isomorphic.

57

Chapter 5. Cartan subalgebras

58

Chapter 6

Classification and the UCT

For A a simple unital C∗-algebra, the Elliot invariant is:

Ell(A) = (K0(A), K0(A)+, [1A]0, K1(A), T (A), rA : T (A)→ S(K0(A))) ,

here T (A) is the trace space and rA the paring rA(τ)([p]) = [τ(p)].

Elliot’s conjecture: Separable, simple, nuclear are classifiable by Elliot’s invariants.

Theorem 6.0.1. Separable, simple, unital, nuclear, Z-stable, UCT algebras are classifi-able by Elliot’s invariants.

An example of a classification theorem: Elliot’s theorem,

Theorem 6.0.2. Let A and B unital AF-algebras and

α : K0(A)→ K1(A)

a unital order isomorphism, i.e.

α(K0(A)+) ⊆ K0(B)+ and α([1A]) = [1B].

Then there exists a unital ∗-isomorphism φ;A→ B such that φ∗ = α.

59

Chapter 6. Classification and the UCT

60

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