- 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


Outline of the talk

Introduction

The Larson-Sweedler theorem

Locally compact quantum groups

The Haar weights and modular theory

Conclusions

References


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.


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.


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.


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 .


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.


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.


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.


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 .


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)).


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.


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‖.


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.


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.


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.


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), ...


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.


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.


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.


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.


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.


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.


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.


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.


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) <∞.


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.


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.


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.


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 )?



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.



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.



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).


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.


References

J. Kustermans & S. Vaes: Locally compact quantum

groups. Ann. Sci. Éc. Norm. Sup. (2000).

J. Kustermans & S. Vaes: Locally compact quantum

groups in the von Neumann algebra. Math. Scand. (2003).

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].

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