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))n2N 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))n2N 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 L1(G) { a von Neumann algebra equipped with the coproduct 1 1 1 ∆ : 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 2 B(L2(G) ⊗ L2(G)) { the multiplicative unitary associated to G: ∗ 1 ∆(f ) = W G(f ⊗ 1)(W G) ; f 2 L (G): Note the inclusions 1 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 L1(G) { a von Neumann algebra equipped with the coproduct 1 1 1 ∆ : 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 2 B(L2(G) ⊗ L2(G)) { the multiplicative unitary