Haagerup Property for Locally Compact and Classical Quantum Groups Based on Joint Work with M

Haagerup property for locally compact and classical quantum groups based on joint work with M. Daws, P. Fima and S. White

Adam Skalski

IMPAN and University of Warsaw

2nd Seminar on Harmonic Analysis and Application 5-7 January 2013, IPM, Tehran

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 1 / 30 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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 2 / 30 Equivalent deﬁnitions of the Haagerup property

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

G admits a mixing unitary representation which weakly contains the trivial representation; there exists a normalised sequence of continuous, positive definite functions on G vanishing at infinity which converges uniformly to 1 on compact subsets of G; there exists a proper, continuous conditionally negative definite function on G; G admits a proper continuous affine action on a real Hilbert space.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 3 / 30 Equivalent deﬁnitions of the Haagerup property

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 3 / 30 Equivalent deﬁnitions of the Haagerup property

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 3 / 30 Equivalent deﬁnitions of the Haagerup property

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 3 / 30 General notations G – a locally compact quantum group `ala 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)) u C0(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

∞ 00 C0(G) ⊂ Cb(G) ⊂ L (G) = C0(G)

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 4 / 30 General notations G – a locally compact quantum group `ala 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)) u C0(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

∞ 00 C0(G) ⊂ Cb(G) ⊂ L (G) = C0(G)

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 4 / 30 General notations G – a locally compact quantum group `ala 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)) u C0(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

∞ 00 C0(G) ⊂ Cb(G) ⊂ L (G) = C0(G)

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 4 / 30 General notations G – a locally compact quantum group `ala 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)) u C0(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

∞ 00 C0(G) ⊂ Cb(G) ⊂ L (G) = C0(G)

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 4 / 30 General notations G – a locally compact quantum group `ala 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)) u C0(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

∞ 00 C0(G) ⊂ Cb(G) ⊂ L (G) = C0(G)

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 4 / 30 Dual quantum groups

Each LCQG G admits the dual LCQG Gb.

∞ 2 L (Gb), C0(Gb) – subalgebras of B(L (G))

W G ∈ M(C0(Gb) ⊗ C0(G)) and

W Gb = (σ(W G))∗

In particular for G – locally compact group

L∞(Gb) = VN(G) ∗ u ∗ C0(Gb) = Cr (G), C0 (Gb) = C (G)

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 5 / 30 Dual quantum groups

Each LCQG G admits the dual LCQG Gb.

∞ 2 L (Gb), C0(Gb) – subalgebras of B(L (G))

W G ∈ M(C0(Gb) ⊗ C0(G)) and

W Gb = (σ(W G))∗

In particular for G – locally compact group

L∞(Gb) = VN(G) ∗ u ∗ C0(Gb) = Cr (G), C0 (Gb) = C (G)

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 5 / 30 Dual quantum groups

Each LCQG G admits the dual LCQG Gb.

∞ 2 L (Gb), C0(Gb) – subalgebras of B(L (G))

W G ∈ M(C0(Gb) ⊗ C0(G)) and

W Gb = (σ(W G))∗

In particular for G – locally compact group

L∞(Gb) = VN(G) ∗ u ∗ C0(Gb) = Cr (G), C0 (Gb) = C (G)

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 5 / 30 Dual quantum groups

Each LCQG G admits the dual LCQG Gb.

∞ 2 L (Gb), C0(Gb) – subalgebras of B(L (G))

W G ∈ M(C0(Gb) ⊗ C0(G)) and

W Gb = (σ(W G))∗

In particular for G – locally compact group

L∞(Gb) = VN(G) ∗ u ∗ C0(Gb) = Cr (G), C0 (Gb) = C (G)

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 5 / 30 Further properties of LCQGs

Deﬁnition A locally compact quantum groups G is

compact if C0(G) is unital (equivalently the Haar weights are ﬁnite); discrete if Gˆ is compact; unimodular if the left and right Haar weights coincide;

amenable if L∞(G) admits a bi-invariant mean; u coamenable if the universal and reduced algebras C0(G) and C0 (G) are naturally isomorphic.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 6 / 30 Further properties of LCQGs

Deﬁnition A locally compact quantum groups G is

compact if C0(G) is unital (equivalently the Haar weights are ﬁnite); discrete if Gˆ is compact; unimodular if the left and right Haar weights coincide;

amenable if L∞(G) admits a bi-invariant mean; u coamenable if the universal and reduced algebras C0(G) and C0 (G) are naturally isomorphic.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 6 / 30 Further properties of LCQGs

Deﬁnition A locally compact quantum groups G is

compact if C0(G) is unital (equivalently the Haar weights are ﬁnite); discrete if Gˆ is compact; unimodular if the left and right Haar weights coincide;

amenable if L∞(G) admits a bi-invariant mean; u coamenable if the universal and reduced algebras C0(G) and C0 (G) are naturally isomorphic.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 6 / 30 Further properties of LCQGs

Deﬁnition A locally compact quantum groups G is

compact if C0(G) is unital (equivalently the Haar weights are ﬁnite); discrete if Gˆ is compact; unimodular if the left and right Haar weights coincide;

amenable if L∞(G) admits a bi-invariant mean; u coamenable if the universal and reduced algebras C0(G) and C0 (G) are naturally isomorphic.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 6 / 30 Further properties of LCQGs

Deﬁnition A locally compact quantum groups G is

compact if C0(G) is unital (equivalently the Haar weights are ﬁnite); discrete if Gˆ is compact; unimodular if the left and right Haar weights coincide;

amenable if L∞(G) admits a bi-invariant mean; u coamenable if the universal and reduced algebras C0(G) and C0 (G) are naturally isomorphic.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 6 / 30 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), quantum ax + b, Eq(2) (amenable and coamenable); + quantum symmetry groups: quantum permutation groups Sn , quantum + automorphism groups of Wang Gaut(Mn), quantum orthogonal groups On (mostly non-coamenable).

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 7 / 30 Some examples of locally compact quantum groups

locally compact groups (all coamenable); duals of locally compact groups (all amenable);

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 7 / 30 Some examples of locally compact quantum groups

locally compact groups (all coamenable); duals of locally compact groups (all amenable);

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 7 / 30 Some examples of locally compact quantum groups

locally compact groups (all coamenable); duals of locally compact groups (all amenable);

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 7 / 30 Representations of LCQGs

Deﬁnition A (unitary) representation of G on a Hilbert space H is a unitary U ∈ 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 weakly contains the trivial representation/has almost invariant vectors if there exists a net 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 of G on L2(G); it is mixing.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 8 / 30 Representations of LCQGs

Deﬁnition A (unitary) representation of G on a Hilbert space H is a unitary U ∈ 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

U(a ⊗ ξi ) − a ⊗ ξi −→ 0.

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 8 / 30 Deﬁnition via representations

Deﬁnition A locally compact quantum group G has HAP if it admits a mixing representation weakly containing the trivial representation.

Proposition

If Gˆ is coamenable, then G has HAP. In particular, amenable discrete quantum groups 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 of the left regular representation.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 9 / 30 Deﬁnition via representations

Deﬁnition A locally compact quantum group G has HAP if it admits a mixing representation weakly containing the trivial representation.

Proposition

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 9 / 30 Deﬁnition via representations

Deﬁnition A locally compact quantum group G has HAP if it admits a mixing representation weakly containing the trivial representation.

Proposition

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 9 / 30 Deﬁnition via representations

Deﬁnition A locally compact quantum group G has HAP if it admits a mixing representation weakly containing the trivial representation.

Proposition

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 9 / 30 ‘Typicality’ of mixing representations

Assume that C0(G) is separable (‘G is second countable’) and ﬁx an inﬁnite dimensional separable Hilbert space H. Then Rep (H) is a Polish space with a G natural (‘point-weak’ topology). Theorem A locally compact quantum group G has HAP if and only if the set of mixing representations is dense in Rep (H). G The proof follows that in the classical case.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 10 / 30 Definition via ‘positive definite functions’ What should continuous positive definite functions on G be? There are at least two possible points of view u via Bochner’s theorem, states on C0(Gb); ∞ elements in Cb(G) yielding ‘completely positive multipliers’ on L (Gb). For a study of related notions see a recent paper of M.Daws and P.Salmi in Journal of Functional Analysis. Theorem Let G be a locally compact quantum group. Then the following conditions are equivalent:

G has HAP; u there exists a net of states (µi )i∈I on C0 (Gb) 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 completely positive multipliers which forms an approximate identity for C0(G).

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 11 / 30 Definition via ‘positive definite functions’ What should continuous positive definite functions on G be? There are at least two possible points of view u via Bochner’s theorem, states on C0(Gb); ∞ elements in Cb(G) yielding ‘completely positive multipliers’ on L (Gb). For a study of related notions see a recent paper of M.Daws and P.Salmi in Journal of Functional Analysis. Theorem Let G be a locally compact quantum group. Then the following conditions are equivalent:

G has HAP; u there exists a net of states (µi )i∈I on C0 (Gb) 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 completely positive multipliers which forms an approximate identity for C0(G).

G has HAP; u there exists a net of states (µi )i∈I on C0 (Gb) 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 completely positive multipliers which forms an approximate identity for C0(G).

G has HAP; u there exists a net of states (µi )i∈I on C0 (Gb) 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 completely positive multipliers which forms an approximate identity for C0(G).

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 11 / 30 Corollary for subgroups

Let G, H be locally compact quantum groups. We say that H a closed quantum subgroup of G in the sense of Woronowicz if there is a surjective ∗ u u -homomorphism π : C0 (G) → C0 (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 (in the sense of Woronowicz), then H has HAP.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 12 / 30 Cnd functions, and convolution semigroups of states From now on we pass to discrete quantum groups. Classical Schönberg correspondence says that the conditionally negative definite functions are ‘generators of families of positive definite functions’:

ψ ←→ exp(−tψ).

Let G be a discrete quantum group. Then the algebra C(Gb) contains a natural dense Hopf *-algebra, Pol(Gb). States on Pol(Gb) are in one-to-one u correspondence with the states on C (Gb). Deﬁnition

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

i µt+s = µt ? µs := (µt ⊗ µs ) ◦ ∆ˆ , t, s ≥ 0; G t→0+ ii µt (a) −→ µ0(a) := (a), a ∈ Pol(Gb).

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 13 / 30 Cnd functions, and convolution semigroups of states From now on we pass to discrete quantum groups. Classical Schönberg correspondence says that the conditionally negative definite functions are ‘generators of families of positive definite functions’:

ψ ←→ exp(−tψ).

Let G be a discrete quantum group. Then the algebra C(Gb) contains a natural dense Hopf *-algebra, Pol(Gb). States on Pol(Gb) are in one-to-one u correspondence with the states on C (Gb). Deﬁnition

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

i µt+s = µt ? µs := (µt ⊗ µs ) ◦ ∆ˆ , t, s ≥ 0; G t→0+ ii µt (a) −→ µ0(a) := (a), a ∈ Pol(Gb).

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 13 / 30 Conditionally negative deﬁnite functions versus generating functionals

The following theorem is essentially due to M. Sch¨urmann. Theorem (Quantum Sch¨onberg correspondence)

Each convolution semigroup of states on Pol(Gb) possesses a generating functional L : Pol(Gb) → C:

µt (a) − (a) L(a) = lim , a ∈ Pol(Gˆ ). t→0+ t The functional L is selfadjoint, vanishes at 1 and is positive on the kernel of the counit; in turn each functional enjoying these properties generates a convolution semigroup of states.

Thus – conditionally negative functions on G correspond to generating functionals on Pol(Gb).

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 14 / 30 Conditionally negative deﬁnite functions versus generating functionals

The following theorem is essentially due to M. Sch¨urmann. Theorem (Quantum Sch¨onberg correspondence)

Each convolution semigroup of states on Pol(Gb) possesses a generating functional L : Pol(Gb) → C:

Thus – conditionally negative functions on G correspond to generating functionals on Pol(Gb).

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 14 / 30 Deﬁnition via conditionally negative deﬁnite functions

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

i G has HAP; u ii there exists a convolution semigroup of states (µt )t≥0 on C0(Gb) such that each at := (id ⊗ µt )(W ) is an element of c0(G), and at tend strictly to 1 as t → 0+;

iii Gb admits a symmetric proper generating functional.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 15 / 30 Deﬁnition via proper aﬃne actions on Hilbert spaces?

There is one remaining ‘classical’ equivalent definitions of HAP – it is the one related 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 in the affine way...

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 16 / 30 Deﬁnition via proper cocycles

G - discrete quantum group.

Unitary representations of G on H correspond to unital ∗-representations of Pol(Gb) on H. Deﬁnition If π : Pol(Gb) → B(H) is a unital ∗-homomorphism, then we say that c : Pol(Gb) → H is a cocycle for π if

c(ab) = π(a)c(b) + c(a)(b), a, b ∈ Pol(Gˆ ) The next theorem is inspired by the ideas introduced by R.Vergnioux. Theorem A discrete quantum group G has HAP if and only if it admits a proper real cocycle.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 17 / 30 Deﬁnition via proper cocycles

G - discrete quantum group.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 17 / 30 HAP via the approximation property for the von Neumann algebra

Recall that a vNa M with a faithful normal tracial state τ has the von Neumann algebraic Haagerup approximation property if there exists a family of unital completely positive τ-preserving normal maps (Φi )i∈I on M such that each of the 2 respective induced maps Ti on L (M, τ) is compact and for each x ∈ M

i∈I Φi (x) −→ 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 algebraic Haagerup approximation property.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 18 / 30 HAP via the approximation property for the von Neumann algebra

i∈I Φi (x) −→ x

σ-weakly.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 18 / 30 vNa HAP in the quantum context Theorem Let G be a discrete unimodular quantum group. Then G has HAP if and only if L∞(Gˆ ) has the von Neumann algebraic Haagerup approximation property.

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

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 19 / 30 vNa HAP in the quantum context Theorem Let G be a discrete unimodular quantum group. Then G has HAP if and only if L∞(Gˆ ) has the von Neumann algebraic Haagerup approximation property.

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 19 / 30 Examples

The following (non-amenable) discrete quantum groups are now known to have HAP: + + duals of quantum orthogonal and unitary groups, OcN , UcN (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 Neumann algebraic Haagerup approximation property.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 20 / 30 Examples

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 20 / 30 Further examples

The last two examples are even more recent:

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

SU^q(1, 1) (M.Caspers, 2013) Here HAP is proved via the construction of suitable completely positive multipliers. The second is not discrete.

Go to summary

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 21 / 30 Further examples

The last two examples are even more recent:

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

SU^q(1, 1) (M.Caspers, 2013) Here HAP is proved via the construction of suitable completely positive multipliers. The second is not discrete.

Go to summary

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 21 / 30 Free product of discrete quantum groups

Deﬁnition

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

C(G\1 ∗ G2) = C(Gb 1) ? C(Gb 2) (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 also ∞ ∞ ∞ L (G\1 ∗ G2) = L (Gb 1) ? L (Gb 2) (von Neumann algebraic free product, with respect to Haar states).

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 22 / 30 Free product of discrete quantum groups

Deﬁnition

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

C(G\1 ∗ G2) = C(Gb 1) ? C(Gb 2) (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 also ∞ ∞ ∞ L (G\1 ∗ G2) = L (Gb 1) ? L (Gb 2) (von Neumann algebraic free product, with respect to Haar states).

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 22 / 30 Free product of discrete quantum groups

Deﬁnition

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

C(G\1 ∗ G2) = C(Gb 1) ? C(Gb 2) (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 also ∞ ∞ ∞ L (G\1 ∗ G2) = L (Gb 1) ? L (Gb 2) (von Neumann algebraic free product, with respect to Haar states).

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 22 / 30 Easy case - unimodular discrete quantum groups

Theorem

Let G1, G2 be discrete unimodular quantum groups with HAP. Then G1 ? G2 has HAP.

Proof.

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 23 / 30 Easy case - unimodular discrete quantum groups

Theorem

Let G1, G2 be discrete unimodular quantum groups with HAP. Then G1 ? G2 has HAP.

Proof.

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 23 / 30 Free products of the convolution semigroups of states

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

Let G1, G2 be discrete quantum groups, with respective convolution semigroups of states (φt )t≥0 and (ωt )t≥0 on Pol(Gc1) and Pol(Gc2). Then (φt ωt )t≥0 is a convolution semigroup of states on Pol(Gb), where G = G1 ∗ G2. The symbol denotes here the conditionally free product of states `ala Boca or Bo˙zejko (with respect to the Haar states on Gc1 and Gc2). Theorem

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 24 / 30 Free products of the convolution semigroups of states

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

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 24 / 30 Summary

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

G admits a mixing representation weakly containing the trivial representation; the set of mixing representations is dense in Rep (H); G c0(G) admits an approximate unit built of ‘positive deﬁnite functions’; Gˆ admits a symmetric proper generating functional; G admits a real proper cocycle; L∞(Gˆ ) has the von Neumann algebraic Haagerup approximation property.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 25 / 30 Open questions

how to characterise the HAP for discrete non-unimodular G via the von Neumann algebra L∞(Gb)? how to deﬁne the Haagerup approximation property for a von Neumann algebra with a state? (in fact we know the answer now!) what are the right versions of the conditionally negative deﬁnite function / cocycle characterization of HAP for non-discrete locally compact quantum groups?

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 26 / 30 Open questions

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 26 / 30 Open questions

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 26 / 30 Open questions

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

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 26 / 30 Property (T ) for quantum groups

Property (T) was studied mainly for discrete quantum groups by P.Fima, D.Kyed and P.Soltan. Here are some facts:

a discrete quantum group with Property (T) is unimodular and finitely generated (shown by Fima); Property (T) admits a characterisation in terms of positive definite functions and cocycles (in the discrete case due to Kyed; the first one is valid also in the locally compact case, as shown by M.Daws, S.White and AS) there exist various incarnations of Kazhdan pairs

and some questions what are interesting examples of locally compact quantum groups with Property (T)? what are the connections between Property (T) for locally compact quantum groups and weakly mixing representations?

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 27 / 30 Property (T ) for quantum groups

Property (T) was studied mainly for discrete quantum groups by P.Fima, D.Kyed and P.Soltan. Here are some facts:

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 27 / 30 Property (T ) for quantum groups

Property (T) was studied mainly for discrete quantum groups by P.Fima, D.Kyed and P.Soltan. Here are some facts:

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 27 / 30 References

Classical Haagerup property

U. Haagerup, An example of a nonnuclear C ∗-algebra, which has the metric approximation property, Invent. Math., 1979. P.A. Cherix, M. Cowling, P. Jolissaint, P. Julg and A. Valette, “Groups with the Haagerup property. Gromov’s a-T-menability”, 2001.

Locally compact quantum groups J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. Ecole´ Norm. Sup., 2000.

HAP for locally compact quantum groups:

M.Daws, P.Fima, A.S. and S.White, Haagerup property for locally compact quantum groups, J. Reine Angew. Math., to appear.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 28 / 30 References

Other recent literature on the Haagerup property for locally compact quantum groups

M.Brannan, Approximation properties for free orthogonal and free unitary quantum groups, J. Reine Angew. Math., 2012. M. Brannan, Reduced operator algebras of trace-preserving quantum automorphism 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 quantum groups FOF , preprint, 2013. M.Caspers, Weak amenability of locally compact quantum groups and approximation properties of extended quantum SU(1, 1), preprint, 2013.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 29 / 30 References

Property (T ) for locally compact quantum groups

P.Fima, Kazhdan’s property T for discrete quantum groups, Internat. J. Math., 2010 D. Kyed, A cohomological description of property (T) for quantum groups, J. Funct. Anal., 2011. D. Kyed and P. Soltan, Property (T) and exotic quantum group norms, J. Noncommut. Geom., 2012. M.Daws, P.Fima, A.S. and S.White, Notes on Property (T) for locally compact quantum groups, in preparation.

Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 7 January 2013 30 / 30