Haagerup property for locally compact quantum groups joint work with M. Daws, P. Fima and S. White
Adam Skalski
IMPAN and University of Warsaw
Workshop on Operator Spaces, Harmonic Analysis and Quantum Probability 10-14 June 2013, Madrid
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 1 / 28 1 Haagerup property for locally compact groups
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 10-14 June 2013 2 / 28 Origins
The story of the Haagerup property starts (in a sense) from the celebrated 1979 result: Theorem (Haagerup)
The length function l : Fn → N0 on the free group of n-generators is (proper) and conditionally negative definite.
1 In other words, by Sch¨onberg correspondence, the family (γ 7→ exp(− n l(γ))n∈N is a normalised sequence of positive definite functions vanishing at infinity, which converges uniformly to 1 on finite subsets of Fn.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 3 / 28 Origins
The story of the Haagerup property starts (in a sense) from the celebrated 1979 result: Theorem (Haagerup)
The length function l : Fn → N0 on the free group of n-generators is (proper) and conditionally negative definite.
1 In other words, by Sch¨onberg correspondence, the family (γ 7→ exp(− n l(γ))n∈N is a normalised sequence of positive definite functions vanishing at infinity, which converges uniformly to 1 on finite subsets of Fn.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 3 / 28 Equivalent definitions of the Haagerup property
In the current language the relevant property of the free group is called 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.
A representation π : G → B(H) is called mixing if its coefficients are C0-functions on G.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 4 / 28 Equivalent definitions of the Haagerup property
In the current language the relevant property of the free group is called 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.
A representation π : G → B(H) is called mixing if its coefficients are C0-functions on G.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 4 / 28 Equivalent definitions of the Haagerup property
In the current language the relevant property of the free group is called 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.
A representation π : G → B(H) is called mixing if its coefficients are C0-functions on G.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 4 / 28 Equivalent definitions of the Haagerup property
In the current language the relevant property of the free group is called 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.
A representation π : G → B(H) is called mixing if its coefficients are C0-functions on G.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 4 / 28 Equivalent definitions of the Haagerup property
In the current language the relevant property of the free group is called 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.
A representation π : G → B(H) is called mixing if its coefficients are C0-functions on G.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 4 / 28 Basic properties and examples
amenable groups have HAP; free groups, finitely generated Coxeter groups have HAP;
SL(2, Z), SO(1, n) and SU(1, n) have HAP;
if Γ1, Γ2 are discrete and have HAP, then Γ1 ? Γ2 has HAP; if Γ is discrete then it has HAP if and only if VN(Γ) has the von Neumann algebraic Haagerup approximation property; G has both HAP and property (T ) if and only if G is compact.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 5 / 28 Basic properties and examples
amenable groups have HAP; free groups, finitely generated Coxeter groups have HAP;
SL(2, Z), SO(1, n) and SU(1, n) have HAP;
if Γ1, Γ2 are discrete and have HAP, then Γ1 ? Γ2 has HAP; if Γ is discrete then it has HAP if and only if VN(Γ) has the von Neumann algebraic Haagerup approximation property; G has both HAP and property (T ) if and only if G is compact.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 5 / 28 Basic properties and examples
amenable groups have HAP; free groups, finitely generated Coxeter groups have HAP;
SL(2, Z), SO(1, n) and SU(1, n) have HAP;
if Γ1, Γ2 are discrete and have HAP, then Γ1 ? Γ2 has HAP; if Γ is discrete then it has HAP if and only if VN(Γ) has the von Neumann algebraic Haagerup approximation property; G has both HAP and property (T ) if and only if G is compact.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 5 / 28 Basic properties and examples
amenable groups have HAP; free groups, finitely generated Coxeter groups have HAP;
SL(2, Z), SO(1, n) and SU(1, n) have HAP;
if Γ1, Γ2 are discrete and have HAP, then Γ1 ? Γ2 has HAP; if Γ is discrete then it has HAP if and only if VN(Γ) has the von Neumann algebraic Haagerup approximation property; G has both HAP and property (T ) if and only if G is compact.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 5 / 28 Basic properties and examples
amenable groups have HAP; free groups, finitely generated Coxeter groups have HAP;
SL(2, Z), SO(1, n) and SU(1, n) have HAP;
if Γ1, Γ2 are discrete and have HAP, then Γ1 ? Γ2 has HAP; if Γ is discrete then it has HAP if and only if VN(Γ) has the von Neumann algebraic Haagerup approximation property; G has both HAP and property (T ) if and only if G is compact.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 5 / 28 Basic properties and examples
amenable groups have HAP; free groups, finitely generated Coxeter groups have HAP;
SL(2, Z), SO(1, n) and SU(1, n) have HAP;
if Γ1, Γ2 are discrete and have HAP, then Γ1 ? Γ2 has HAP; if Γ is discrete then it has HAP if and only if VN(Γ) has the von Neumann algebraic Haagerup approximation property; G has both HAP and property (T ) if and only if G is compact.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 5 / 28 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 10-14 June 2013 6 / 28 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 10-14 June 2013 6 / 28 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 10-14 June 2013 6 / 28 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 10-14 June 2013 6 / 28 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 10-14 June 2013 6 / 28 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 10-14 June 2013 7 / 28 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 10-14 June 2013 7 / 28 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 10-14 June 2013 7 / 28 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 10-14 June 2013 7 / 28 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 10-14 June 2013 8 / 28 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 10-14 June 2013 8 / 28 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 10-14 June 2013 8 / 28 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 10-14 June 2013 8 / 28 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 10-14 June 2013 8 / 28 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 10-14 June 2013 9 / 28 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 10-14 June 2013 9 / 28 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 10-14 June 2013 9 / 28 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 10-14 June 2013 9 / 28 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 is in fact the left regular representation of G on L2(G); it is mixing.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 10 / 28 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 is in fact the left regular representation of G on L2(G); it is mixing.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 10 / 28 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 ).
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 11 / 28 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 ).
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 11 / 28 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 ).
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 11 / 28 ‘Typicality’ of mixing representations
Assume for the moment that C0(G) is separable (‘G is second countable’) and fix an infinite dimensional separable Hilbert space H. Then Rep (H) is a Polish G space with a 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
This seems not to be recorded in literature even for standard groups, although has likely been known to experts (see work of Halmos, Bergelson+Rosenblatt and Hjorth).
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 12 / 28 ‘Typicality’ of mixing representations
Assume for the moment that C0(G) is separable (‘G is second countable’) and fix an infinite dimensional separable Hilbert space H. Then Rep (H) is a Polish G space with a 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
This seems not to be recorded in literature even for standard groups, although has likely been known to experts (see work of Halmos, Bergelson+Rosenblatt and Hjorth).
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 12 / 28 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 JFA paper of M.Daws and P.Salmi. 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 10-14 June 2013 13 / 28 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 JFA paper of M.Daws and P.Salmi. 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 10-14 June 2013 13 / 28 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 JFA paper of M.Daws and P.Salmi. 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 10-14 June 2013 13 / 28 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 JFA paper of M.Daws and P.Salmi. 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 10-14 June 2013 13 / 28 Corollary for subgroups
Let G, H be locally compact quantum groups. 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 10-14 June 2013 14 / 28 Cnd functions, and convolution semigroups of states From now on we pass to discrete quantum groups. Classical Sch¨onberg 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 10-14 June 2013 15 / 28 Cnd functions, and convolution semigroups of states From now on we pass to discrete quantum groups. Classical Sch¨onberg 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 10-14 June 2013 15 / 28 Conditionally negative definite functions versus generating functionals
The following theorem is essentially due to M. Sch¨urmann. Theorem 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 10-14 June 2013 16 / 28 Conditionally negative definite functions versus generating functionals
The following theorem is essentially due to M. Sch¨urmann. Theorem 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 10-14 June 2013 16 / 28 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 10-14 June 2013 17 / 28 Definition via proper affine actions on Hilbert spaces?
There is one more of the four ‘classical’ equivalent definitions of the Haagerup property – 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 10-14 June 2013 18 / 28 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 10-14 June 2013 19 / 28 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 10-14 June 2013 19 / 28 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 10-14 June 2013 20 / 28 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 10-14 June 2013 20 / 28 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 10-14 June 2013 21 / 28 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 10-14 June 2013 21 / 28 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 10-14 June 2013 21 / 28 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 10-14 June 2013 21 / 28 Some examples
The following (non-amenable) discrete quantum groups are known to have HAP:
+ duals of quantum permutation groups, ScN (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 10-14 June 2013 22 / 28 Some examples
The following (non-amenable) discrete quantum groups are known to have HAP:
+ duals of quantum permutation groups, ScN (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 10-14 June 2013 22 / 28 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).
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 23 / 28 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).
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 23 / 28 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).
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 23 / 28 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 Jolissaint 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 10-14 June 2013 24 / 28 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 Jolissaint 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 10-14 June 2013 24 / 28 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. This allows us to construct examples of non-unimodular non-amenable discrete quantum groups with HAP.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 25 / 28 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. This allows us to construct examples of non-unimodular non-amenable discrete quantum groups with HAP.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 25 / 28 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. This allows us to construct examples of non-unimodular non-amenable discrete quantum groups with HAP.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 25 / 28 Summary
Let G be a discrete, ‘second countable’ 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 10-14 June 2013 26 / 28 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? 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 10-14 June 2013 27 / 28 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? 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 10-14 June 2013 27 / 28 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? 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 10-14 June 2013 27 / 28 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? 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 10-14 June 2013 27 / 28 References Classical Haagerup property 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.
This talk: M.Daws, P.Fima, A.S. and S.White, Haagerup property for locally compact quantum groups, preprint, 2013.
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. F.Lemeux, Haagerup property for quantum reflection groups, preprint, 2013.
Adam Skalski (IMPAN & UW) Haagerup property for LCQGs 10-14 June 2013 28 / 28