- Quantum groups. From the algebraic to the operator ... · Introduction The Larson-Sweedler...
Transcript of - Quantum groups. From the algebraic to the operator ... · Introduction The Larson-Sweedler...
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Quantum groups. From the algebraic to the operatoralgebraic approach
A. Van Daele
Department of MathematicsUniversity of Leuven
November 2014 / NBFAS Leeds
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Outline of the talk
Introduction
The Larson-Sweedler theorem
Locally compact quantum groups
The Haar weights and modular theory
Conclusions
References
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Introduction
First some historical comments:
The operator algebra approach to quantum groups finds itsorigin in the attempts to generalize Pontryagin’s duality forabelian locally compact groups to the case of all locallycompact groups. This generalization started with the workof Tannaka (1938) and Krein (1949).
On the other hand, the theory of quantum groups, asdeveloped by e.g. Drinfel’d, follows a different line ofresearch, in a purely algebraic context, starting with thedevelopment of the notion of a Hopf algebra.
A new boost in both research fields in the late 80’s:
the work of Woronowicz on the quantum SUq(2),
the work of Drinfel’d and Jimbo obtaining the quantizationsof universal enveloping algebras.
However, even then, there was still little interaction.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Hopf algebras
Definition
A Hopf algebra is a pair (A,∆) of an algebra with identity and acoproduct ∆. The coproduct is a unital homomorphism∆ : A → A ⊗ A satisfying coassociativity (∆⊗ ι)∆ = (ι⊗∆)∆.It is assumed that there is a counit ε. This is an algebrahomomorphism ε : A → C satisfying
(ε⊗ ι)∆(a) = a (ι⊗ ε)∆(a) = a
for all a. Also the existence of an antipode is assumed. It is ananti-homomorphism S : A → A satisfying
m(S ⊗ ι)∆(a) = ε(a)1 m(ι⊗ S)∆(a) = ε(a)1
for all a ∈ A. Here m is the multiplication map.
The counit and the antipode are unique if they exist.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
The basic examples
The two basic examples associated with any finite group G:
Example
Consider the algebra K (G) of all complex functions on G.
Define ∆(f )(p, q) = f (pq) for f ∈ K (G) and p, q ∈ G. This is a
coproduct. Define ε(f ) = f (e) where e is the identity in G. This
is the counit. Finally, let S(f )(p) = f (p−1) for p ∈ G, then S is
the antipode.
If G is no longer finite, one can define a multiplier Hopf algebra.
Example
Consider the group algebra CG of G and use p 7→ λp for the
imbedding of G in CG. A coproduct is defined by
∆(λp) = λp ⊗ λp. The counit satisfies ε(λp) = 1 and the
antipode S(λp) = λp−1 for all p.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Locally compact groups
Let G be a locally compact group.
Theorem
Let A be the C∗-algebra C0(G) of complex continuous functions
on G, tending to 0 at infinity. Consider A ⊗ A = C0(G × G) and
its multiplier algebra M(A ⊗ A) = Cb(G × G), the algebra of
bounded continuous complex functions on G × G. If we define
∆ : A → M(A ⊗ A) by
∆(f )(p, q) = f (pq),
then (A,∆) is a locally compact quantum group.
There is a counit as in the finite case. It is given by ε : f 7→ f (e)where e is the identity. Indeed
(ι⊗ ε)∆(f )(p) = f (pe) = f (p) (ε⊗ ι)∆(f )(p) = f (ep) = f .
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Locally compact groups - the antipode
There is also an antipode. It is given by S(f )(p) = f (p−1) forf ∈ C0(G) and p ∈ G. It satisfies
m(ι⊗ S)∆(f )(p) = f (pp−1) = f (e) = ε(f )
m(S ⊗ ι)∆(f )(p) = f (p−1p) = f (e) = ε(f ).
This works because
ι⊗ S and S ⊗ ι are isomorphisms of Cb(G × G),
m : Cb(G × G) → Cb(G) is defined by m(f )(p) = f (p, p),
also f 7→ f (e) is a homomorphism defined on Cb(G).
However, all of this is no longer true in the case of a generallocally compact quantum group where A is no longer abelian.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Integrals on Hopf algebras
Definition
A left integral on a Hopf algebra (A,∆) is a non-zero linearfunctional ϕ : A → C satisfying (ι⊗ ϕ)∆(a) = ϕ(a)1 for all a
(left invariance). A right integral is a non-zero linear functional ψon A satisfying (ψ ⊗ ι)∆(a) = ψ(a)1 for all a (right invariance).
Integrals are unique (up to a scalar) if they exist. On afinite-dimensional Hopf algebra, integrals always exist.
Example
If A = K (G) for a finite group G, the left and right integrals are
the same and given by ϕ(f ) =∑
p∈G f (p). Also for CG the
integrals coincide and are given by ϕ(λp) = 1 if p = e and
ϕ(λp) = 0 otherwise.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
The Larson-Sweedler theorem for Hopf algebras
Theorem
Assume that A is an algebra with identity and a full coproduct
∆. If there is a faithful left integral and a faithful right integral,
then (A,∆) is a Hopf algebra.
A coproduct on a unital algebra is called full if elements ofthe form
(ι⊗ ω)∆(a) (ω ⊗ ι)∆(a),
where a ∈ A and ω is a linear functional on A, each span A.In the original formulation, it is assumed that there is acounit. That is a stronger condition.Compare fullness of the coproduct with the densityconditions we need in the C∗-algebraic formulation.All this is precisely as in the definition of a locally compactquantum group in the operator algebraic framework.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
The proof: the underlying ideas
The following results motivate the proof.
Proposition
Let G be a finite group and f a complex function on G. Write
f (p) = f (pqq−1) =∑
gi(pq)hi(q).
Then∑
gi(q)hi(pq) = f (q.(pq)−1) = f (p−1).
The result is easily generalized to Hopf algebras.
Proposition
Let (A,∆) be a Hopf algebra and a ∈ A. Write
a ⊗ 1 =∑
∆(bi)(1 ⊗ ci).
Then∑
(1 ⊗ bi)∆(ci) = S(a)⊗ 1. Also ε(a)1 =∑
bici .
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Proof of the Larson-Sweedler theorem
Assume that we have a unital algebra A with a full coproduct ∆.Assume that ϕ is a faithful left integral and that ψ is a faithfulright integral on A. Consider p, q ∈ A and write
(ι⊗ ι⊗ ϕ)(∆13(p)∆23(q)) =∑
bi ⊗ ci .
Then∑
∆(bi)(1 ⊗ ci) = (ι⊗ ι⊗ ϕ)((ι⊗∆)(∆(p)(1 ⊗ q))) = a ⊗ 1
where a = (ι⊗ ϕ)(∆(p)(1 ⊗ q)) by the left invariance of ϕ.Similarly we find∑
(1 ⊗ bi)∆(ci) = (ι⊗ ι⊗ ϕ)((ι⊗∆)((1 ⊗ p)∆(q))) = b ⊗ 1
where b = (ι⊗ ϕ)((1 ⊗ p)∆(q)).
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Proof of the Larson-Sweedler theorem - continued
Now we would like to define S : A → A by S(a) = b. In order todo that, we need to verify two things:
A is spanned by elements of the form (ι⊗ ϕ)(∆(p)(1 ⊗ q)).
If∑
∆(bi)(1 ⊗ ci) = 0, then∑
bi ⊗ ci = 0.
For the first result, we need that ϕ is faithful and that ∆ is full.For the second, we need the right integral ψ. Indeed, multiplywith ∆(p) from the left and apply ψ ⊗ ι to get
∑ψ(pbi)ci = 0.
As this is true for all p ∈ A and because ψ is assumed to befaithful, it follows that
∑bi ⊗ ci = 0.
The counit is defined in a similar way by ε(a) = ϕ(pq) ifa = (ι⊗ ϕ)((∆(p)(1 ⊗ q)).
The proof is completed by showing that ε is the counit and thatS is the antipode. This is straightforward.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
C∗-algebraic locally compact quantum groups
Definition (Kustermans and Vaes)
A locally compact quantum group is a pair (A,∆) of a
C∗-algebra A and a full coproduct ∆ on A such that there exists
a left and a right Haar weight.
The coproduct is a non-degenerate ∗-homomorphism∆ : A → M(A ⊗ A) satisfying (∆⊗ ι)∆ = (ι⊗∆)∆.It is full in the sense that the spaces spanned by(ω ⊗ ι)∆(a) and (ι⊗ ω)∆(a) are dense in A.A left Haar weight is a faithful lower semi-continuoussemi-finite central weight ϕ on A satisfying
ϕ((ω ⊗ ι)∆(a)) = ω(1)ϕ(a)
for all positive ω ∈ A∗ and all positive elements a ∈ A
satisfying ϕ(a) <∞ (left invariance). Here ω(1) = ‖ω‖.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Von Neumann algebraic l.c. quantum groups
Definition (Kustermans and Vaes)
A locally compact quantum group is a pair (M,∆) of a von
Neumann algebra M and a coproduct ∆ on M such that there
exists a left and a right Haar weight.
The coproduct is a unital normal ∗-homomorphism∆ : M → M ⊗ M satisfying (∆⊗ ι)∆ = (ι⊗∆)∆.
A left Haar weight is a faithful normal semi-finite weight ϕon A satisfying
ϕ((ω ⊗ ι)∆(x)) = ω(1)ϕ(a)
for all positive ω ∈ M∗ and all positive elements x ∈ M
satisfying ϕ(x) <∞ (left invariance).
Similarly for a right Haar weight ψ.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Locally compact quantum groupsThe basic examples
Let G be any locally compact group.
Theorem
Let M = L∞(G). Define ∆ on M by ∆(f )(p, q) = f (pq)whenever p, q ∈ G. Then (M,∆) is a locally compact quantum
group. The left and right Haar weights are the integrals with
respect to the left and the right Haar measure respectively.
For f ∈ L∞(G) we have
((ι⊗ ϕ)∆(f ))(p) =
∫f (pq)dq =
∫f (q)dq.
In the C∗-framework, one takes A = C0(G) and defines∆ : A → M(A ⊗ A) as above. The Haar weights are now faithfullower semi-continuous, semi-finite (and central) weights.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Locally compact quantum groupsThe basic examples - continued
Theorem
Let M = VN(G), the von Neumann algebra generated by the
left translations λp on L2(G). There is a coproduct ∆ satisfying
∆(λp) = λp ⊗ λp for all p. The pair (M,∆) is also a locally
compact quantum group. The left Haar weight satisfies
ϕ
(∫f (p)λpdp
)= f (e)
when f ∈ Cc(G). Now the left and right Haar weights coincide.
In the C∗-framework, one takes A = C∗r (G), the reduced
C∗-algebra of G. Also here the coproduct is a non-degenerate∗-homomorphism from A → M(A ⊗ A) defined as above. Theleft and right Haar weights are still the same and they are givenby the same formula as in the von Neumann case.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Locally compact quantum groupsMore history and other comments
Pontryagin duality is a symmetric theory: the dual object G
is again a locally compact group.
This is no longer the case with Tannaka-Krein duality andits generalizations.
The first self-dual theory comes with the Kac algebras: Kacand Vainerman (1973), Enock and Schwartz (1973).
The new developments (Woronowicz’ SUq(2) (1987) andthe Drinfel’d-Jimbo examples (1985)), show that the theoryis too restrictive (the antipode S is assumed to be a ∗-map).
New research by various people finally led to the theory oflocally compact quantum groups as we know it now: Baajand Skandalis (1993), Masuda, Nakagami andWoronowicz (1995), Kustermans and Vaes(1999), ...
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Locally compact quantum groupsMore comments
Consider again the definition.
Definition
A locally compact quantum group is a pair (M,∆) of a vonNeumann algebra M and a coproduct ∆ on M such that thereexists a left and a right Haar weight.
It is nice and simple (if compared e.g. with the definition ofa Kac algebra - and it is more general).There is no counit, nor an antipode from the start - theexistence is proven (at least of the antipode).This is in contrast with (1) the definition of a (locallycompact) group and with (2) the notion of a Hopf algebra.It is assumed that the Haar weights exist whereas in thetheory of locally compact groups, the existence is provenfrom the axioms.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
The Haar weights
Differences between the algebraic and the operator algebraictheory:
The combination of the algebra and the coproduct is quitecomplicated, even in the setting of Hopf algebras.
In the operator algebraic approach, more complicationsare a consequence of working with completed tensorproducts, an unbounded counit and antipode, weights, ...
In general, the counit is not considered. And the antipodehas a nice polar decomposition that makes it tractable.
Using weights on C∗-algebras and von Neumann algebrasinvolve the modular theory.
We will now further concentrate on some aspects of the weightsthat arise in the theory of locally compact quantum groups.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Weights on C∗-algebras
Definition
Let A be a C∗-algebra. A weight on a C∗-algebra is a mapϕ : A+ → [0,∞] satisfying
ϕ(λa) = λϕ(a) for all a ∈ A+ and λ ∈ R with λ ≥ 0,
ϕ(a + b) = ϕ(a) + ϕ(b) for all a, b ∈ A+.
Given such a weight we have two associated sets.
NotationWe denote
Nϕ = {a ∈ A | ϕ(a∗a) <∞}
and set Mϕ = N∗ϕNϕ, the subspace of A spanned by elements
of the form a∗b with a, b ∈ Nϕ.
Clearly Nϕ is a left ideal and Mϕ is a ∗-subalgebra.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
The GNS representation
Proposition
Assume that the weight is lower semi-continuous. Let F be the
set of positive linear functionals ω on A, majorized by ϕ on A+.
Then for all positive elements a we have:
ϕ(a) = supω∈F ω(a)
Assume further that the weight ϕ is semi-finite, lowersemi-continuous and faithful.
Proposition
There is a non-degenerate faithful representation πϕ on a
Hilbert space Hϕ and a linear map Λϕ : Nϕ → Hϕ such that
ϕ(b∗a) = 〈Λϕ(a),Λϕ(b)〉
and such that Λϕ(ab) = πϕ(a)Λϕ(b) for a ∈ A and b ∈ Nϕ.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
The GNS representation
Proposition
For all ω majorized by ϕ there is a vector ξ in Hϕ and a
bounded operator π′(ξ) in the commutant πϕ(A)′ satisfying
π′(ξ)Λϕ(a) = πϕ(a)ξ and ω(a) = 〈πϕ(a)ξ, ξ〉.
Such a vector is called right bounded.
Proposition
Consider the closure of the subspace of Hϕ spanned by the
right bounded vectors. This space is invariant under the
commutant πϕ(A)′. Denote by p the projection onto this space.
Then p ∈ πϕ(A)′′.
Remark that it is possible that p 6= 1.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Extension of weights
Denote by A the double dual A∗∗ of A. Any element ω ∈ A∗ hasa unique normal extension ω to A.
Proposition
Let ϕ be a lower semi-continuous semi-finite weight on A.
Define ϕ : A+ → [0,∞] by
ϕ(x) = supω∈F ω(x).
Then ϕ is a normal, semi-finite weight on A.
A weight on a von Neumann algebra is defined as a weight ona C∗-algebra. It is said to be normal if ϕ(x) = sup ϕ(xα) for anyincreasing net xα → x .
The additivity is the non-trivial part. The GNS space of theextension is the same as for the original weight.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Central weights on C∗-algebras
Notation
We denote by e the support of ϕ. It is the largest projection
in A such that ϕ(1 − e) = 0.
Consider also the unique normal extension of πϕ to A. It is
a normal ∗-homomorphism from A onto πϕ(A)′′. Denote by
f the support projection of this extension. It belongs to the
center of A.
Proposition
πϕ(e) = p (projection on the right bounded vectors),
f is the central support of e in A.
Definition
The weight ϕ is called central if its support e is central in A.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Central weights on C∗-algebrasRemarks
Let us make the following remarks.
Remark
Central weights ϕ on a C∗-algebra A are precisely the ones
that have a faithful extension to πϕ(A)′′.
In the work of Kustermans and Vaes, they are called
approximately KMS weights, and the density of the right
bounded vectors is used as the definition.
In the work of Masuda, Nakagami and Woronowicz, still
another characterization is used.
Such weights precisely allow the use of the modular theory
and left Hilbert algebra techniques.
Also uniqueness of Haar weights is treated more easily fromthis point of view.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Locally compact quantum groups - Recall
Definition
Let A be a C∗-algebra with a coproduct ∆ : A → M(A ⊗ A).Assume that elements of the form
(ω ⊗ ι)∆(a) and (ι⊗ ω)∆(a)
each span a dense subspace of A. Assume that there is a leftHaar weight ϕ and a right Haar weight ψ on (A,∆). Then thispair is called a locally compact quantum group.
A coproduct ∆ on A is a non-degenerate ∗-homomorphismfrom A to M(A ⊗ A) where the minimal tensor product is taken.It is assumed to be coassociative: (∆⊗ ι)∆ = (ι⊗∆)∆.A left Haar weight is a proper central weight ϕ that is leftinvariant:
ϕ((ω ⊗ ι)∆(a)) = ω(1)ϕ(a)
for all positive ω in A∗ and all a ∈ A+ such that ϕ(a) <∞.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Extension to A
Proposition
The coproduct extends to a unital and normal ∗-homomorphism
∆ : A → A ⊗ A (where now we consider the von Neumann
tensor product). The normal extensions ϕ and ψ are still left,
respectively right invariant weights on (A, ∆).
The extension of the coproduct is standard. We have a unitalimbedding of M(A ⊗ A) in A ⊗ A and hence a unique normalextension ∆ : A → A ⊗ A. Coassociativity on this level is aconsequence of the uniqueness of normal extensions.
The left invariance of the extension ϕ is less trivial. Themodular theory of weights is used as the main tool. Similarly forthe right invariance of ψ.
Important remark: The extended weights ϕ and ψ are (ingeneral) no longer faithful.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Uniqueness of the supports
Let e and f be the supports of ϕ and ψ. They are centralprojections in A by assumption.
Proposition
The support projections e and f are equal. Hence they are
independent of the choices of Haar weights ϕ and ψ on (A,∆).
Proof (sketch): From the left invariance of ϕ, it follows that∆(1 − e)(1 ⊗ (1 − e)) = 0. Multiply with ∆(x) where x ∈ A+
and ψ(x) <∞. Then apply right invariance of ψ to arrive atψ(x(1 − e)) = 0. This implies that f (1 − e) = 0. Similarly wefind e(1 − f ) = 0 and so e = f .
Remark
It is this result that makes it possible to pass, from the very
beginning, to the study of locally compact quantum groups in
the von Neumann algebra framework.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Restriction to the von Neumann algebra πϕ(A)′′
Now we restrict the coproduct and the Haar weights again.
Theorem
Denote M = Ae and define ∆0 : M → M ⊗ M by
∆0(x) = ∆(x)(e ⊗ e). Then ∆0 is a coproduct on M.The
restrictions ϕ0 and ψ0 of the weights ϕ and ψ are now faithful
invariant functionals. Hence, the pair (M,∆0) is a locally
compact quantum group in the von Neumann algebraic setting.
Most of this is now straightforward. One needs the equality∆(e)(1 ⊗ e) = (1 ⊗ e) to obtain that ∆0 is still unital.
Also observe that M is isomorphic with πϕ(A)′′ because we cutdown to the support of πϕ in A.
This completes the passage from the C∗-algebra to the vonNeumann algebra approach. The other direction is standard.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
The formula ϕ(δ12 · δ
12 )
For any locally compact quantum group (M,∆) one can provethe following. It gives the relation of the left Haar weight ϕ withthe right Haar weight ψ by means of the modular element.
Theorem
There is a unique, non-singular positive self-adjoint operator δ,
affiliated with M, such that ψ = ϕ(δ12 · δ
12 ). It satisfies
σt(δ) = ν tδ where (σt) are the modular automorphisms
associated with ϕ and where ν is the scaling constant.
In the case of a Hopf algebra, the scaling constant is defined byϕ ◦ S2 = νϕ. In the case of locally compact quantum groups, asimilar formula holds.
The question here is: How to give a meaning to the formulaϕ(δ
12 · δ
12 )?
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
The formula ϕ(δ12 · δ
12 )
We start with a von Neumann algebra M and a faithful normalsemi-finite weight ϕ on M. Moreover we assume that (δis)s∈R isa strongly continuous one-parameter group of unitaries in M
satisfying σt(δis) = ν istδis for all s, t where ν is a strictly positive
real number.
As before, we consider the left ideal Nϕ of elements x ∈ M
satisfying ϕ(x∗x) <∞ and the canonical map Λϕ : Nϕ → Hϕ ofNϕ into the GNS space Hϕ. We let M act directly on its GNSspace.
We will define the new weight by first defining N and the mapΛ : N → Hϕ. If these objects have the right properties, they giverise to a faithful normal semi-finite weight on M via a standardprocedure.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
The formula ϕ(δ12 · δ
12 )
Definition
Let N be the space of elements x ∈ M so that xδ12 is bounded
and belongs to Nϕ. Define Λ : N → Hϕ by Λ(x) = Λϕ(xδ12 ).
Proposition
The space N is a dense left ideal of M and the image Λ(N) is
dense in Hϕ.
Different aspects of the proof:
We get enough elements x so that xδ12 is bounded by
taking x∫
f (s)δis ds with appropriate functions f .
Next we construct elements of the form above with x ∈ Nϕ
and∫
f (s)δis ds analytic w.r.t. (σt).
This is possible because σt(δis) = ν istδis.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
The formula ϕ(δ12 · δ
12 )
Theorem
The map Λ : N → Hϕ is closable for the strong topology on M
and the norm topology on Hϕ. Denote the closure as
Λ : N → Hϕ. There is a faithful, normal semi-finite weight ϕ
defined on M by ϕ(x∗x) = ‖Λ(x)‖2 when x ∈ N and
ϕ(x∗x) = ∞ if not.
Different aspects of the proof:
Assume that (xα) is a net in N so that xα → 0 andΛ(xα) → η ∈ Hϕ. With an appropriate choice of right
bounded vectors ξ we have π′(ξ)Λϕ(xαδ12 ) = xαδ
12 ξ → 0 so
that π′(ξ)η = 0 for enough right bounded vectors. Thisimplies η = 0.
Additivity of the weight (standard technique).
Faithfulness and normality of the weight (easy).
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
Conclusions
We talked about two ’interacting’ fields of research: (1) Theoperator algebra approach and (2) the Hopf algebraapproach to quantum groups.In this talk we gave an important example to illustrate theintimate link between the two topics. It is theLarson-Sweedler theorem, as known in Hopf algebratheory, that is implicit in the definition of a quantum groupin the operator algebra setting.In the second part of the talk, we discussed some technicalaspects of the operator algebra approach to the theory. Acrucial role is played here by the modular theory ofweights.A challenge is to generalize all of this to locally compactquantum groupoids. A purely algebraic version exists andis well understood. The same is true for a von Neumannalgebraic theory but not for the C∗-algebraic version of it.
Introduction The Larson-Sweedler theorem Locally compact quantum groups Haar weights and modular theory Conclusions
References
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J. Kustermans & S. Vaes: Locally compact quantum
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R.G. Larson & M.E. Sweedler: An associative orthogonal
bilinear form for Hopf algebras. Amer. J. Math. (1969).
A. Van Daele: Locally compact quantum groups. A von
Neumann algebra approach. Sigma 10 (2014),082,41pages. Arxiv: 0602.212 [math.OA].
A. Van Daele & Shuanhong Wang: The Larson-Sweedler
theorem for multiplier Hopf algebras. J. of Alg. (2006).
B.-J. Kahng & A. Van Daele: The Larson-Sweedler
theorem for weak multiplier Hopf algebras. Arxiv:1406.0299 [math.RA].