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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: 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: 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: 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 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 7 January 2013 4 / 30 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 7 January 2013 4 / 30 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 7 January 2013 4 / 30 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 7 January 2013 4 / 30 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 7 January 2013 4 / 30 Dual quantum groups Each LCQG G admits the dual LCQG Gb. 1 2 L (Gb), C0(Gb) { subalgebras of B(L (G)) W G 2 M(C0(Gb) ⊗ C0(G)) and W Gb = (σ(W G))∗ In particular for G { locally compact group L1(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. 1 2 L (Gb), C0(Gb) { subalgebras of B(L (G)) W G 2 M(C0(Gb) ⊗ C0(G)) and W Gb = (σ(W G))∗ In particular for G { locally compact group L1(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. 1 2 L (Gb), C0(Gb) { subalgebras of B(L (G)) W G 2 M(C0(Gb) ⊗ C0(G)) and W Gb = (σ(W G))∗ In particular for G { locally compact group L1(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. 1 2 L (Gb), C0(Gb) { subalgebras of B(L (G)) W G 2 M(C0(Gb) ⊗ C0(G)) and W Gb = (σ(W G))∗ In particular for G { locally compact group L1(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 G^ is compact; unimodular if the left and right Haar weights coincide; amenable if L1(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 G^ is compact; unimodular if the left and right Haar weights coincide; amenable if L1(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 G^ is compact; unimodular if the left and right Haar weights coincide; amenable if L1(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 G^ is compact; unimodular if the left and right Haar weights coincide; amenable if L1(G) admits a bi-invariant mean; u coamenable if the universal and reduced algebras C0(G) and C0 (G) are naturally isomorphic.
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