Property (T) and Haagerup property for quantumgroups – a global point of view

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

Adam Skalski

IMPAN

Brazos Analysis Seminar25 March 2017

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 1 / 21

Unitary representationsG – locally compact group

A (unitary, strongly continuous) representation π of G on a Hilbert space H

contains an invariant vector if

∃ξ∈H,‖ξ‖=1 ∀g∈G π(g)ξ = ξ;

contains almost invariant vectors if

∃ξi∈H,‖ξi‖=1 ∀g∈G π(g)ξi − ξi −→ 0

(uniformly on compact subsets)

is mixing if∀ξ,η∈H 〈ξ, π(·)η〉 ∈ C0(G );

is ergodic if it does not contain an invariant vector;

is weakly mixing if π ⊗ π is ergodic.

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 2 / 21

Haagerup property and property (T)

Definition

G has the Haagerup property (HAP) if it admits a mixing representation withalmost invariant vectors.

Definition

G has the Kazhdan Property (T) if its every representation with almost invariantvectors contains an invariant vector.

Amenable groups have HAP; G has both HAP and property (T ) if and only if Gis compact

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 3 / 21

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

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

∆(f ) = WG(f ⊗ 1)(WG)∗, f ∈ L∞(G).

Note the inclusions

C0(G) ⊂ Cb(G) ⊂ L∞(G) = C0(G)′′

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 4 / 21

Dual quantum groups

Each LCQG G admits the dual LCQG G.

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

WG ∈ M(C0(G)⊗ C0(G)) and

W G = (σ(WG))∗

In particular for G – locally compact group

L∞(G ) = VN(G )

C0(G ) = C∗r (G ), C u0 (G) = C∗(G )

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 5 / 21

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;

of Kac type if the so-called scaling group is trivial (the antipode is abounded map);

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

coamenable if the universal and reduced algebras C0(G) and Cu0(G) are

naturally isomorphic;

second countable if C0(G) is separable.

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 6 / 21

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, usually not Kac);

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

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

(mostly non-coamenable, mostly Kac).

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 7 / 21

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 ofU.

Representations of G are in a 1-1 correpondence with C∗-representations ofCu0(G).

One can also tensor representations of G (U >V ), take direct sums (U ⊕ V )and pass to a contragredient representation Uc .

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 8 / 21

Representations of LCQGs – continued

Definition

A representation U of G is mixing if all its coefficients belong to C0(G). It hasalmost invariant vectors if there exists a net of unit vectors (ξi )i∈I such that forall a ∈ C0(G)

U(a⊗ ξi )− a⊗ ξi −→ 0

– equivalently for all b ∈ Cu0(G)

φU(b)ξi − ε(b)ξi −→ 0 strictly.

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

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 9 / 21

Definitions and first facts

Definition

A locally compact quantum group G has the Haagerup property (HAP) if itadmits a mixing representation containing almost invariant vectors.

Definition

A locally compact quantum group G has Kazhdan Property (T) if its everyrepresentation containing almost invariant vectors contains an invariant vector.

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).

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 10 / 21

Space of representations

G – second countable locally compact quantum group, H – fixed infinitedimensional separable Hilbert space. Then RepG(H) is a Polish space with anatural (‘point-weak’) topology.It is equipped with two natural operations: direct sum (finite or countable) andtensoring (after we fix some unitary identifications of H⊗ H with H, etc.).

Lemma (DFSW)

Suppose R ⊂ RepG(H)

i is stable under unitary equivalence;

ii is stable under tensoring with any V ∈ RepG(H);

iii contains a representation with almost invariant vectors.

Then R is dense in RepG(H).

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 11 / 21

HAP and density of mixing representations

Theorem (DFSW)

A second countable locally compact quantum group G has HAP if and only if theset of mixing representations is dense in RepG(H).

Proof.⇐=Approximate the trivial representation with mixing ones and take their direct sum(still mixing!).=⇒Use the lemma with R – mixing representations.

Ideas go back to the work of Halmos for Z.

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 12 / 21

(T) via ‘typical’ representations

Theorem (Kerr-Pichot, 2012)

Let G – classical locally compact group, second countable. Then G does not haveProperty (T) if and only if weakly mixing representations are dense in RepG (H).

We want to show the same for (a class of) quantum groups.Recall: U ∈ RepG(H) is weakly mixing if U >Uc is ergodic.

when is the class of weakly mixing representations stable under tensoring?

when not (T) means that there is a weakly mixing representation withalmost invariant vectors?

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 13 / 21

Weak mixing representations revisited

Classically: U is not weakly mixing if and only if U >Uc contains an invariantvector if and only if U contains a finite dimensional subrepresentation.More generally: U >V c contains a fixed vector if and only if U and V contain thesame finite dimensional subrepresentation.

Lemma (Chen+Ng 2015, see also Kyed+So ltan, Viselter)

If G is of Kac type, then a representation of G is weakly mixing if and only if itdoes not contain a finite dimensional subrepresentation; hence then the class ofweakly mixing representations is stable under tensoring.

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 14 / 21

Weak mixing representations with almost invariant vectors

Let us contradict the statement: there is a representation of G which is weaklymixing and has almost invariant vectors.

Definition

G has Property (T)1,1 (of Bekka and Valette) if for every representation U of Gwith almost invariant vectors U >Uc has a fixed vector.

Obviously (T)=⇒ (T)1,1.

Theorem (Bekka and Valette)

For classical groups (T)⇐⇒ (T)1,1.

This gives the result of Kerr and Pichot.

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 15 / 21

Discrete quantum groups with low duals

� – discrete quantum group. Then

c0(�) =⊕i∈I

Mni

Definition� as above has a low dual if supi∈I ni <∞.

In other words, we have a uniform bound on the size of irreducible representationsof the compact quantum group dual to �.

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 16 / 21

Main theorem

Theorem (DSV, 2016)

Let � – discrete unimodular second countable quantum group with a low dual.Then � has Property (T) if and only if it has Property (T)1,1

Corollary (DSV, 2016)

Let � – discrete unimodular second countable quantum group with a low dual.Then � does not have (T) if and only if weakly mixing representations form adense Gδ-set in Rep�(H).

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 17 / 21

Main theorem – ingredients of the proof

Problem: assume � does not have (T). Construct a representation U of � withalmost invariant vectors such that U >Uc contains no invariant vector.

Idea (Jolissaint): use non (T) of � to construct a semigroup of states on the

algebra C(�) with particular properties

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 18 / 21

Main theorem – ingredients of the proofActual ingredients:

development of the notion of Kazhdan pairs for quantum groups withProperty (T);

application of this to showing that if a locally compact G is of Kac type anddoes not have (T) then one can find a net of positive-definite normalisedpositive elements in Cb(G) which converge to 1 strictly, but not in norm;

use of Yukio Arano’s work on central Property (T) to show that for discreteunimodular case one can choose the elements above in the centre;

construction of a strongly unbounded symmetric generating functional L on�;

using L to generate a convolution semigroup of states µt on Cu(�);

building out of µt ‘symmetric’ GNS representations of Cu(�) (thusself-contragredient representations Ut of �);

showing that the representations Ut >Ut cannot contain invariant vectors(and only here the low dual assumption comes in), and concluding byanother Kazhdan pair argument.

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 19 / 21

Other consequences of (T)⇐⇒(T)1,1

Theorem (DSV, 2016)

Let � – discrete unimodular second countable quantum group with a low dual.Then the following are equivalent:

i � has Property (T);

ii for every action of � on a von Neumann algebra invariant states are limits ofnormal invariant states;

iii every ergodic action of � on a von Neumann algebra preserving a faithfulnormal state is strongly operator ergodic (i.e. asymptotically invariant netsof elements in the von Neumann algebra are trivial).

These generalize classical results of Li, Ng, Connes and Weiss.

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 20 / 21

References

M.Daws, P.Fima, A.S. and S.White, Haagerup property for locally compactquantum groups, Crelle, 2016.

M.Daws, A.S. and A.Viselter, Around Property (T) for locally compact quantumgroups, Comm.Math.Phys., to appear.

Adam Skalski (IMPAN) HAP and (T) – global viewpoint Brazos 21 / 21