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Haagerup property for locally compact and classicalquantum groups

based on joint work with M. Daws, P. Fima and S. White

Adam Skalski

IMPAN and University of Warsaw

8th Jikji workshop19-23 August 2013, NIMS, Daejeon

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 19-23 August 2013 1 / 29

1 Haagerup property for locally compact groups recalled

2 Locally compact quantum groups – general facts

3 Haagerup property for locally compact quantum groups

4 Haagerup property for discrete quantum groups

5 Free products of discrete quantum groups

6 Summary and open questions


Equivalent definitions of the Haagerup property

A locally compact group G has the Haagerup property (HAP) if the followingequivalent properties hold:

G admits a mixing unitary representation which weakly contains the trivialrepresentation;

there exists a normalised sequence of continuous, positive definite functionson G vanishing at infinity which converges uniformly to 1 on compactsubsets of G ;

there exists a proper, continuous conditionally negative definite function onG ;

G admits a proper continuous affine action on a real Hilbert space.


General notationsG – a locally compact quantum group a la Kustermans-Vaes

L∞(G) – a von Neumann algebra equipped with the coproduct

∆ : L∞(G)→ L∞(G)⊗L∞(G)

carrying all the information about G

C0(G) – the corresponding (reduced) C∗-object, Cb(G) := M(C0(G))

Cu0(G) – the universal version of C0(G)

L2(G) – the GNS Hilbert space of the right invariant Haar weight on G

W G ∈ B(L2(G)⊗ L2(G)) – the multiplicative unitary associated to G:

∆(f ) = W G(f ⊗ 1)(W G)∗, f ∈ L∞(G).

Note the inclusions

C0(G) ⊂ Cb(G) ⊂ L∞(G) = C0(G)′′


Dual quantum groups

Each LCQG G admits the dual LCQG G.

L∞(G), C0(G) – subalgebras of B(L2(G))

W G ∈ M(C0(G)⊗ C0(G)) and

W G = (σ(W G))∗

In particular for G – locally compact group

L∞(G ) = VN(G )

C0(G ) = C∗r (G ), C u0 (G) = C∗(G )


Further properties of LCQGs

DefinitionA locally compact quantum groups G is

compact if C0(G) is unital (equivalently the Haar weights are finite);

discrete if G is compact;

unimodular if the left and right Haar weights coincide;

amenable if L∞(G) admits a bi-invariant mean;

coamenable if the universal and reduced algebras C0(G) and C u0 (G) are

naturally isomorphic.


Some examples of locally compact quantum groups

locally compact groups (all coamenable);

duals of locally compact groups (all amenable);

quantum deformations of classical Lie groups: for example SUq(2), quantumax + b, Eq(2) (amenable and coamenable);

quantum symmetry groups: quantum permutation groups S+n , quantum

automorphism groups of Wang Gaut(Mn), quantum orthogonal groups O+n

(mostly non-coamenable).


Representations of LCQGs

Definition

A (unitary) representation of G on a Hilbert space H is a unitaryU ∈ M(C0(G)⊗ K (H)) such that

(∆⊗ ι)(U) = U13U23.

The operators (ι⊗ωξ,η)(U) ∈ Cb(G), where ξ, η ∈ H, are called coefficients of U.

Definition

A representation U of G is mixing if all its coefficients belong to C0(G). It weaklycontains the trivial representation/has almost invariant vectors if there exists anet of unit vectors (ξi )i∈I such that for all a ∈ C0(G)

U(a⊗ ξi )− a⊗ ξi −→ 0.

The multiplicative unitary W G plays the role of the left regular representation ofG on L2(G); it is mixing.


Definition via representations

DefinitionA locally compact quantum group G has HAP if it admits a mixing representationweakly containing the trivial representation.

Proposition

If G is coamenable, then G has HAP. In particular, amenable discrete quantumgroups have HAP. G is compact if and only if it has both HAP and Property (T ).

This follows as coamenability is equivalent to the weak containment property ofthe left regular representation.


‘Typicality’ of mixing representations

Assume that C0(G) is separable (‘G is second countable’) and fix an infinitedimensional separable Hilbert space H. Then RepG(H) is a Polish space with anatural (‘point-weak’ topology).

TheoremA locally compact quantum group G has HAP if and only if the set of mixingrepresentations is dense in RepG(H).

The proof follows that in the classical case.


Definition via ‘positive definite functions’What should continuous positive definite functions on G be? There are at leasttwo possible points of view

via Bochner’s theorem, states on Cu0(G);

elements in Cb(G) yielding ‘completely positive multipliers’ on L∞(G).

For a study of related notions see a recent paper of M.Daws and P.Salmi inJournal of Functional Analysis.

TheoremLet G be a locally compact quantum group. Then the following conditions areequivalent:

G has HAP;

there exists a net of states (µi )i∈I on C u0 (G) such that the net

((id⊗ µi )(W ))i∈I is an approximate identity in C0(G);

there is a net (ai )i∈I in C0(G) of representing elements of completelypositive multipliers which forms an approximate identity for C0(G).


Corollary for subgroups

Let G, H be locally compact quantum groups. We say that H a closed quantumsubgroup of G in the sense of Woronowicz if there is a surjective∗-homomorphism π : C u

0 (G)→ C u0 (H) such that

(π ⊗ π) ◦∆G = ∆H ◦ π.

Corollary

If G has HAP and is coamenable, and H is a closed quantum subgroup of G (inthe sense of Woronowicz), then H has HAP.


Cnd functions, and convolution semigroups of states

From now on we pass to discrete quantum groups.

Classical Schonberg correspondence says that the conditionally negative definitefunctions are ‘generators of families of positive definite functions’:

ψ ←→ exp(−tψ).

Let G be a discrete quantum group. Then the algebra C(G) contains a natural

dense Hopf *-algebra, Pol(G). States on Pol(G) are in one-to-one

correspondence with the states on Cu(G).

Definition

A convolution semigroup of states on Pol(G) is a family (µt)t≥0 of states on

Pol(G) such that

i µt+s = µt ? µs := (µt ⊗ µs) ◦∆G, t, s ≥ 0;

ii µt(a)t→0+−→ µ0(a) := ε(a), a ∈ Pol(G).


Conditionally negative definite functions versus generatingfunctionals

The following theorem is essentially due to M. Schurmann.

Theorem (Quantum Schonberg correspondence)

Each convolution semigroup of states on Pol(G) possesses a generating functional

L : Pol(G)→ C:

L(a) = limt→0+

µt(a)− ε(a)

t, a ∈ Pol(G).

The functional L is selfadjoint, vanishes at 1 and is positive on the kernel of thecounit; in turn each functional enjoying these properties generates a convolutionsemigroup of states.

Thus – conditionally negative functions on G correspond to generatingfunctionals on Pol(G).


Definition via conditionally negative definite functions

TheoremLet G be a discrete quantum group. The following are equivalent:

i G has HAP;

ii there exists a convolution semigroup of states (µt)t≥0 on Cu0(G) such that

each at := (id⊗ µt)(W ) is an element of c0(G), and at tend strictly to 1 ast → 0+;

iii G admits a symmetric proper generating functional.


Definition via proper affine actions on Hilbert spaces?

There is one remaining ‘classical’ equivalent definitions of HAP – it is the onerelated to proper affine actions of G on Hilbert spaces.But we do not even quite know what it means that G acts on a Hilbert space inthe affine way...

We thus need to view affine isometric actions simply as cocycles for unitaryrepresentations.


Definition via proper cocycles

G - discrete quantum group.

Unitary representations of G on H correspond to unital ∗-representations ofPol(G) on H.

Definition

If π : Pol(G)→ B(H) is a unital ∗-homomorphism, then we say that

c : Pol(G)→ H is a cocycle for π if

c(ab) = π(a)c(b) + c(a)ε(b), a, b ∈ Pol(G)

The next theorem is inspired by the ideas introduced by R.Vergnioux.

TheoremA discrete quantum group G has HAP if and only if it admits a proper realcocycle.


HAP via the approximation property for the von Neumannalgebra

Recall that a vNa M with a faithful normal tracial state τ has the von Neumannalgebraic Haagerup approximation property if there exists a family of unitalcompletely positive τ -preserving normal maps (Φi )i∈I on M such that each of therespective induced maps Ti on L2(M, τ) is compact and for each x ∈ M

Φi (x)i∈I−→ x

σ-weakly.

P.Jolissaint showed that this property does not depend on the choice of τ .

Theorem (M. Choda)

A discrete group Γ has HAP if and only if VN(Γ) has the von Neumann algebraicHaagerup approximation property.


vNa HAP in the quantum context

TheoremLet G be a discrete unimodular quantum group. Then G has HAP if and only ifL∞(G) has the von Neumann algebraic Haagerup approximation property.

Proof.Follows the classical idea of Choda: if G has HAP, we have good positive definitefunctions, so constructing multipliers out of them (see M.Junge + M.Neufang +Z.J.Ruan, later also M.Daws) yields the approximation property for L∞(G) (thisdoes not use the unimodularity).The other direction is based on ‘averaging’ approximating maps on L∞(G) intomultipliers. Here unimodularity seems crucial.

Corollary

Let G be a discrete unimodular quantum group. Then G has HAP if and only ifC(G) has (a stronger version of) the C∗-algebraic Haagerup approximationproperty of Dong.


Examples

The following (non-amenable) discrete quantum groups are now known to haveHAP:

duals of quantum orthogonal and unitary groups, O+N , U+

N (M.Brannan,2012);

duals of Wang’s quantum automorphism groups (M.Brannan, 2013);

duals of quantum reflection groups (F.Lemeux, 2013).

All these cases are unimodular, and the HAP is proved via the von Neumannalgebraic Haagerup approximation property.


Further examples

The last two examples are even more recent:

duals of Wang’s ‘free orthogonal quantum groups’ OF (K.De Commer,A.Freslon, M.Yamashita, 2013);

˜SUq(1, 1) (M.Caspers, 2013)

Here HAP is proved via the construction of suitable completely positivemultipliers. The second is not discrete.

Go to summary


Free product of discrete quantum groups

DefinitionLet G1, G2 be discrete quantum groups. The free product of G1 and G2 is thediscrete quantum group G1 ∗G2 ‘defined’ by the equality

C(G1 ∗G2) = C(G1) ? C(G2)

(universal C∗-product with amalgamation over units).

The last formula is of course inspired by the equality

C∗(Γ1 ∗ Γ2) ≈ C∗(Γ1) ? C∗(Γ2).

We have alsoL∞(G1 ∗G2) = L∞(G1) ? L∞(G2)

(von Neumann algebraic free product, with respect to Haar states).


Easy case - unimodular discrete quantum groups

TheoremLet G1, G2 be discrete unimodular quantum groups with HAP. Then G1 ?G2 hasHAP.

Proof.Note that G1 ?G2 is unimodular and use the result of F.Boca showing that thefree product of von Neumann algebras with the Haagerup approximation propertyhas the Haagerup approximation property.


Free products of the convolution semigroups of states

If G1, G2 are discrete quantum groups with HAP, but are not unimodular, wecannot use the von Neumann characterization.

LemmaLet G1, G2 be discrete quantum groups, with respective convolution semigroups

of states (φt)t≥0 and (ωt)t≥0 on Pol(G1) and Pol(G2). Then (φt � ωt)t≥0 is a

convolution semigroup of states on Pol(G), where G = G1 ∗G2.

The symbol � denotes here the conditionally free product of states a la Boca or

Bozejko (with respect to the Haar states on G1 and G2).

TheoremLet G1, G2 be discrete quantum groups with HAP. Then G1 ∗G2 has HAP.


Summary

Let G be a discrete, ‘second countable’ unimodular quantum group. Thefollowing conditions are equivalent (and can be used as the definition of HAP):

G admits a mixing representation weakly containing the trivialrepresentation;

the set of mixing representations is dense in RepG(H);

c0(G) admits an approximate unit built of ‘positive definite functions’;

G admits a symmetric proper generating functional;

G admits a real proper cocycle;

L∞(G) has the von Neumann algebraic Haagerup approximation property.


Open questions

how to characterise the HAP for discrete non-unimodular G via the vonNeumann algebra L∞(G)?

how to define the Haagerup approximation property for a von Neumannalgebra with a state?

what are the right versions of the conditionally negative definite function /cocycle characterization of HAP for non-discrete locally compact quantumgroups?

can one characterise HAP for G via existence of suitable actions of G onsome C∗-algebras?


References

Classical Haagerup propertyU. Haagerup, An example of a nonnuclear C∗-algebra, which has the metricapproximation property, Invent. Math., 1979.

P.A. Cherix, M. Cowling, P. Jolissaint, P. Julg and A. Valette, “Groups with theHaagerup property. Gromov’s a-T-menability”, 2001.

Locally compact quantum groupsJ. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. EcoleNorm. Sup., 2000.

HAP for locally compact quantum groups:M.Daws, P.Fima, A.S. and S.White, Haagerup property for locally compactquantum groups, preprint, 2013.


References

Other recent literature on the Haagerup property for locally compactquantum groupsM.Brannan, Approximation properties for free orthogonal and free unitaryquantum groups, J. Reine Angew. Math., 2012.

M. Brannan, Reduced operator algebras of trace-preserving quantumautomorphism groups, preprint, 2012.

A. Freslon, Examples of weakly amenable discrete quantum groups, preprint,2012.

F.Lemeux, Haagerup property for quantum reflection groups, preprint, 2013.

K.De Commer, A.Freslon and M.Yamashita, CCAP for the discrete quantumgroups FOF , preprint, 2013.

M.Caspers, Weak amenability of locally compact quantum groups andapproximation properties of extended quantum SU(1, 1), preprint, 2013.


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