Haagerup property for arbitrary von Neumann algebras Martijn - PowerPoint PPT Presentation

Haagerup property for arbitrary von Neumann algebras Martijn Caspers (WWU Münster) joint with Adam Skalski (IMPAN/Warsaw University) related to work by R. Okayasu, R. Tomatsu June 13, 2014

Equivalent notions of the Haagerup property Introduction HAP for von Neumann algebras A group G has the Haagerup property if: HAP for There exists a net of positive definite normalized functions in C 0 ( G ) arbitrary von Neumann converging to 1 uniformly on compacta algebras G admits a proper affine action on a Hilbert space Equivalent notions There exists a proper, conditionally negative function on G Quantum groups

Examples Introduction HAP for von Neumann algebras HAP for Amenable groups arbitrary von F n (Haagerup, ’78/’79) Neumann algebras SL ( 2 , Z ) Equivalent Haagerup property + Property (T) implies compactness notions Quantum groups

HAP for von Neumann algebras Introduction Definition Haagerup property (Choda ’83, Jolissaint ’02) HAP for von Neumann A finite von Neumann algebra ( M , τ ) has HAP if there exists a net (Φ i ) i of normal algebras cp maps Φ i : M → M such that: HAP for τ ◦ Φ i ≤ τ arbitrary von Neumann The map T i : x Ω τ �→ Φ i ( x )Ω τ is compact algebras T i → 1 strongly Equivalent notions Quantum groups Remark: In the definition ( M , τ ) has HAP than Φ i ’s can be chosen unital and such that τ ◦ Φ i = τ .

HAP for groups versus HAP for vNA’s Introduction HAP for von Theorem (Choda ’83) Neumann algebras A discrete group G has HAP ⇔ The group von Neumann algebra L ( G ) has HAP HAP for arbitrary von Neumann Idea of the proof: (Haagerup) algebras Equivalent ⇒ ϕ i the positive definite functions ⇒ Φ i : L ( G ) → L ( G ) : λ ( f ) �→ λ ( ϕ i f ) . notions ⇐ Φ i cp maps ⇒ use the ‘averaging technique’: Quantum groups ϕ i ( s ) = τ ( λ ( s ) ∗ Φ i ( λ ( s )) .

HAP for von Neumann algebras Introduction HAP for von Neumann algebras Definition Haagerup property HAP for A σ -finite von Neumann algebra ( M , ϕ ) has HAP if there exists a net (Φ i ) i of arbitrary von normal cp maps Φ i : M → M such that: Neumann algebras ϕ ◦ Φ i ≤ ϕ Equivalent The map T i : x Ω ϕ �→ Φ i ( x )Ω ϕ is compact notions T i → 1 strongly Quantum groups

HAP for von Neumann algebras Introduction HAP for von Definition Haagerup property (MC, Skalski) Neumann algebras An arbitrary von Neumann algebra ( M , ϕ ) with nsf weight ϕ has HAP if there HAP for exists a net (Φ i ) i of normal cp maps Φ i : M → M such that: arbitrary von Neumann ϕ ◦ Φ i ≤ ϕ algebras The map T i : Λ ϕ ( x ) �→ Λ ϕ (Φ i ( x )) is compact Equivalent notions T i → 1 strongly Quantum groups Remark: In our approach it is essential to treat weights instead of states.

Motivating examples Introduction HAP for von Neumann Brannan ’12: Free orthogonal and free unitary quantum groups have HAP . algebras Kac case ⇒ Semi-finite. HAP for arbitrary von Neumann algebras De Commer, Freslon, Yamashita ’13: Equivalent Non-Kac case of this result ⇒ Non-semi-finite. notions Quantum groups Houdayer, Ricard ’11: Free Araki-Woods factors.

Problems arising? Introduction Definition Haagerup property HAP for von Neumann An arbitrary von Neumann algebra ( M , ϕ ) with nsf weight ϕ has HAP if there algebras exists a net (Φ i ) i of normal cp maps Φ i : M → M such that: HAP for ϕ ◦ Φ i ≤ ϕ arbitrary von Neumann The map T i : Λ ϕ ( x ) �→ Λ ϕ (Φ i ( x )) is compact algebras T i → 1 strongly Equivalent notions Questions: Quantum groups Does the definition depend on the choice of the weight? Can the maps Φ i be taken ucp and ϕ -preserving? Can we always assume that Φ i ◦ σ ϕ t = σ ϕ t ◦ Φ i ?

Weight independence Theorem (MC, A. Skalski) The HAP is independent of the choice of the n.s.f. weight: ( M , ϕ ) has HAP iff Introduction ( M , ψ ) has HAP . HAP for von Neumann algebras Idea of the proof: HAP for arbitrary von Neumann algebras Equivalent notions Quantum groups

Weight independence Theorem (MC, A. Skalski) The HAP is independent of the choice of the n.s.f. weight: ( M , ϕ ) has HAP iff Introduction ( M , ψ ) has HAP . HAP for von Neumann algebras Idea of the proof: HAP for Treat the semi-finite case using Radon-Nikodym derivatives. arbitrary von Neumann algebras ϕ ( h · h ) = ψ ( · ) Equivalent notions Let ϕ have cp maps Φ i . Then formally, Quantum groups i ( · ) := h − 1 Φ i ( h · h ) h − 1 , Φ ′ will yield the cp maps for ψ .

Weight independence Theorem (MC, A. Skalski) The HAP is independent of the choice of the n.s.f. weight: ( M , ϕ ) has HAP iff Introduction ( M , ψ ) has HAP . HAP for von Neumann algebras Idea of the proof: HAP for Treat the semi-finite case using Radon-Nikodym derivatives. arbitrary von Neumann algebras ϕ ( h · h ) = ψ ( · ) Equivalent notions Let ϕ have cp maps Φ i . Then formally, Quantum groups i ( · ) := h − 1 Φ i ( h · h ) h − 1 , Φ ′ will yield the cp maps for ψ . Let α be any ϕ -preserving action of R on ( M , ϕ ) . If ( M ⋊ R , ˆ ϕ ) has HAP then ( M , ϕ ) has HAP .

Weight independence Theorem (MC, A. Skalski) The HAP is independent of the choice of the n.s.f. weight: ( M , ϕ ) has HAP iff Introduction ( M , ψ ) has HAP . HAP for von Neumann algebras Idea of the proof: HAP for Treat the semi-finite case using Radon-Nikodym derivatives. arbitrary von Neumann algebras ϕ ( h · h ) = ψ ( · ) Equivalent notions Let ϕ have cp maps Φ i . Then formally, Quantum groups i ( · ) := h − 1 Φ i ( h · h ) h − 1 , Φ ′ will yield the cp maps for ψ . Let α be any ϕ -preserving action of R on ( M , ϕ ) . If ( M ⋊ R , ˆ ϕ ) has HAP then ( M , ϕ ) has HAP . Use crossed product duality to conclude the converse.

Weight independence Theorem (MC, A. Skalski) The HAP is independent of the choice of the n.s.f. weight: ( M , ϕ ) has HAP iff Introduction ( M , ψ ) has HAP . HAP for von Neumann algebras Idea of the proof: HAP for Treat the semi-finite case using Radon-Nikodym derivatives. arbitrary von Neumann algebras ϕ ( h · h ) = ψ ( · ) Equivalent notions Let ϕ have cp maps Φ i . Then formally, Quantum groups i ( · ) := h − 1 Φ i ( h · h ) h − 1 , Φ ′ will yield the cp maps for ψ . Let α be any ϕ -preserving action of R on ( M , ϕ ) . If ( M ⋊ R , ˆ ϕ ) has HAP then ( M , ϕ ) has HAP . Use crossed product duality to conclude the converse. Conclude from the semi-finite case (Step 1).

Crossed products Introduction HAP for von Neumann Consequence algebras Let α be any action of a group G on M . HAP for If M ⋊ α G has HAP then so has M arbitrary von Neumann If M has HAP and G amenable then M ⋊ α G has HAP algebras Equivalent notions Quantum groups

Crossed products Introduction HAP for von Neumann Consequence algebras Let α be any action of a group G on M . HAP for If M ⋊ α G has HAP then so has M arbitrary von Neumann If M has HAP and G amenable then M ⋊ α G has HAP algebras Equivalent Comments: notions M ⋊ α G has HAP implies that G has HAP in case G discrete Quantum groups Z 2 ⋊ SL ( 2 , Z ) does not have HAP whereas SL ( 2 , Z ) has HAP and is weakly amenable

Markov property Introduction Let M be a von Neumann algebra with normal state ϕ . We say that a normal map HAP for von Neumann Φ : M → M is Markov if it is a ucp ϕ -preserving map. algebras Theorem (MC, A. Skalski) HAP for arbitrary von The following are equivalent: Neumann algebras ( M , ϕ ) has HAP Equivalent ( M , ϕ ) has HAP and the cp maps Φ i are Markov notions Quantum groups Corollary: If ( M 1 , ϕ 1 ) and ( M 2 , ϕ 2 ) have HAP then so does the free product ( M 1 ⋆ M 2 , ϕ 1 ⋆ ϕ 2 ) . (following Boca ’93).

Modular HAP Introduction HAP for von Neumann We say that ( M , ϕ ) has the modular HAP if the cp maps Φ i commute with algebras σ t , t ∈ R . HAP for arbitrary von Theorem (MC, Skalski) Neumann algebras ( M , ϕ ) is the von Neumann algebra of a compact quantum group with Haar state ϕ . TFAE: Equivalent notions ( M , ϕ ) has HAP Quantum ( M , ϕ ) has the modular HAP groups

Introduction HAP for von Questions: Neumann algebras Does the definition depend on the choice of the weight? NO HAP for arbitrary von Can the maps Φ i be taken ucp and ϕ -preserving (Markov)? YES if ϕ is a Neumann state. algebras Can we always assume that Φ i ◦ σ ϕ t = σ ϕ t ◦ Φ i ? YES in every known Equivalent example. notions Quantum groups

Introduction HAP for von Questions: Neumann algebras Does the definition depend on the choice of the weight? NO HAP for arbitrary von Can the maps Φ i be taken ucp and ϕ -preserving (Markov)? YES if ϕ is a Neumann state. algebras Can we always assume that Φ i ◦ σ ϕ t = σ ϕ t ◦ Φ i ? YES in every known Equivalent example. notions Quantum groups Question: Can we find Markov maps in case ( B ( H ) , Tr ) ?