Ozawa’s class S for locally compact groups and unique prime factorization of group von Neumann algebras Tobe Deprez Skyline Communications IPAM, Lake arrowhead, 2019 Tobe Deprez Class S for locally compact groups 1 / 27
Group von Neumann algebra Group von Neumann Group G algebra L ( G ) ◮ Consider left-regular representation λ : G → B � L 2 ( G ) � ( λ g ξ )( h ) = ξ ( g − 1 h ) g , h ∈ Γ, ξ ∈ L 2 ( G ) Group algebra C G = span { λ g } g ∈ G Definition The group von Neumann algebra L ( G ) is the von Neumann algebra generated by C G , i.e. w.o. = span { λ g } g ∈ G w.o. L ( G ) = C G Tobe Deprez Class S for locally compact groups 2 / 27
Problem setting Group von Neumann Group G algebra L ( G ) Question How much does L ( G ) “remember” of the structure of G? ◮ (Connes, 1976) All L ( G ) are isomorphic for G countable, amenable, icc ◮ Open problem : is L ( F n ) ∼ = L ( F m ) if n � = m ? ◮ Ozawa’s class S ◮ G countable: (Ozawa, 2004) , (Ozawa-Popa, 2004) , ... ◮ G locally compact: this talk Tobe Deprez Class S for locally compact groups 3 / 27
Contents 1 Class S for countable groups Definition Applications 2 Topological amenability 3 Class S for locally compact groups Definition My results Tobe Deprez Class S for locally compact groups 4 / 27
Class S for countable groups Definition Contents 1 Class S for countable groups Definition Applications 2 Topological amenability 3 Class S for locally compact groups Definition My results Tobe Deprez Class S for locally compact groups 5 / 27
Class S for countable groups Definition Class S for countable groups Γ countable group Definition (Ozawa, 2006) Γ is in class S (or is bi-exact ) if Γ is exact and ∃ map η : Γ → Prob(Γ) satisfying � η ( gkh ) − g · η ( k ) � → 0 if k → ∞ Examples ◮ Free groups F n k kh kh k η ( k ) = unif. measure on path e to k ◮ Right invariance � η ( kh ) = unif. measure path e to kh η ( k ) = unif. measure path e to k difference: path from k to kh Tobe Deprez Class S for locally compact groups 6 / 27
Class S for countable groups Definition Class S for countable groups Γ countable group Definition (Ozawa, 2006) Γ is in class S (or is bi-exact ) if Γ is exact and ∃ map η : Γ → Prob(Γ) satisfying � η ( gkh ) − g · η ( k ) � → 0 if k → ∞ Examples ◮ Free groups F n g gk gk η ( k ) = unif. measure on path e to k ◮ Left equivariance � η ( gk ) = unif. measure path e to gk g · η ( k ) = unif. measure path g to gk difference: path from e to g Tobe Deprez Class S for locally compact groups 6 / 27
Class S for countable groups Definition Class S for countable groups Γ countable group Definition (Ozawa, 2006) Γ is in class S (or is bi-exact ) if Γ is exact and ∃ map η : Γ → Prob(Γ) satisfying � η ( gkh ) − g · η ( k ) � → 0 if k → ∞ Examples ◮ Free groups F n ◮ Amenable groups ◮ ∃ sequence µ n ∈ Prob(Γ) � µ n − g · µ n � → 0 g gk gkh e ◮ Define | 2 k | η ( k ) = 1 � µ i | k | i = | k | +1 Tobe Deprez Class S for locally compact groups 6 / 27
Class S for countable groups Definition Class S for countable groups Γ countable group Definition (Ozawa, 2006) Γ is in class S (or is bi-exact ) if Γ is exact and ∃ map η : Γ → Prob(Γ) satisfying � η ( gkh ) − g · η ( k ) � → 0 if k → ∞ Examples ◮ Free groups F n ◮ Amenable groups ◮ (Adams, 1994) Hyperbolic groups ◮ (Skandalis, 1988) Lattices in finite-center, connected, simple Lie groups with real rank 1 Tobe Deprez Class S for locally compact groups 6 / 27
Class S for countable groups Definition Exactness Definition (Kirchberg-Wasserman, 1999) Γ is exact if for every short exact sequence 0 → A → B → C → 0 of Γ- C ∗ -algebras, also 0 → A ⋊ r Γ → B ⋊ r Γ → C ⋊ r Γ → 0 is exact. Examples ◮ Almost every group is exact e.g. amenable groups, hyperbolic groups, linear groups, countable subgroups of connected simple Lie groups, ... ◮ Examples of non-exact groups: (Gromov, 2003) and (Osajda, 2014) Tobe Deprez Class S for locally compact groups 7 / 27
Class S for countable groups Applications Contents 1 Class S for countable groups Definition Applications 2 Topological amenability 3 Class S for locally compact groups Definition My results Tobe Deprez Class S for locally compact groups 8 / 27
Class S for countable groups Applications Applications Group von Neumann Group G algebra L ( G ) Theorem (Ozawa, 2004) L (Γ) is solid if Γ is in class S , i.e. for every diffuse N ⊆ L (Γ) von Neumann subalgebra, the algebra N ′ ∩ L (Γ) is amenable. Corollary L (Γ) is prime if Γ is non-amenable, icc and in class S , i.e. L (Γ) �∼ = M 1 ⊗ M 2 if M 1 , M 2 non-type I factors. L ( F 2 × F 2 ) = L ( F 2 ) ⊗ L ( F 2 ) �∼ = L ( F 2 ). Tobe Deprez Class S for locally compact groups 9 / 27
Class S for countable groups Applications Applications Group von Neumann Group G algebra L ( G ) Theorem (Ozawa-Popa, 2004) Let Γ = Γ 1 × · · · × Γ n with Γ i non-amenable, icc and in class S . Then L (Γ) = L (Γ 1 ) ⊗ . . . ⊗ L (Γ n ) has unique prime factorization (UPF) , i.e. if L (Γ) = N 1 ⊗ . . . ⊗ N m for prime factors N 1 , . . . , N m , then n = m and N i ∼ = s L (Γ i ) (after relabeling). L ( F 2 × F 2 × F 2 ) �∼ = L ( F 2 × F 2 ). Tobe Deprez Class S for locally compact groups 10 / 27
Topological amenability Contents 1 Class S for countable groups Definition Applications 2 Topological amenability 3 Class S for locally compact groups Definition My results Tobe Deprez Class S for locally compact groups 11 / 27
Topological amenability Topological amenability – Definition ◮ G locally compact and second countable ◮ X compact topological space, G � X continuous Definition (Anantharaman-Delaroche, 1987) G � X is (topologically) amenable if ∃ weakly* continuous maps µ n : X → Prob( G ) such that � µ n ( g · x ) − g · µ n ( x ) � → 0 uniformly on X and on compact sets for g ∈ G . Examples ◮ If X = { x 0 } , then G � X is amenable iff G is amenable ◮ If X discrete and G � X free, then G � X amenable µ n ( x ) = δ x Tobe Deprez Class S for locally compact groups 12 / 27
Topological amenability Topological amenability – Example Definition (Anantharaman-Delaroche, 1987) G � X is (topologically) amenable if ∃ : µ n : X → Prob( G ) of continuous maps such that � µ n ( g · x ) − g · µ ( x ) � → 0 uniformly on X and on compact sets for g ∈ G . g · x g · x Examples ◮ F 2 � boundary of Cayley graph g x µ n ( x ) = unif. measure on first n vertices of path e to x ◮ µ n ( g · x ) = (...) path e to g · x ◮ g · µ n ( x ) = (...) path g to g · x difference: path from e to g Tobe Deprez Class S for locally compact groups 13 / 27
Topological amenability Topological amenability – Example Definition (Anantharaman-Delaroche, 1987) G � X is (topologically) amenable if ∃ : µ n : X → Prob( G ) of continuous maps such that � µ n ( g · x ) − g · µ ( x ) � → 0 uniformly on X and on compact sets for g ∈ G . Examples ◮ F 2 � boundary of Cayley graph ◮ Γ � boundary of Cayley graph for Γ hyperbolic Tobe Deprez Class S for locally compact groups 13 / 27
Topological amenability Characterization of class S Theorem (Ozawa, 2006) A countable group Γ belongs to class S if and only if Γ has amenable action on a boundary that is small at infinity , i.e. ∃ compactification h Γ of Γ such that ◮ Actions by left and right translation extend to actions on h Γ , ◮ Action by right translation is trivial on ν Γ = h Γ \ Γ , ◮ Action by left translation on ν Γ = h Γ \ Γ is topologically amenable. Tobe Deprez Class S for locally compact groups 14 / 27
Topological amenability Link with C ∗ -algebras Consider the following conditions: (i) G � X is amenable (ii) C ( X ) ⋊ G ∼ = C ( X ) ⋊ r G (iii) C ( X ) ⋊ r G is nuclear Theorem (Anantharaman-Delaroche, 1987) For G countable, we have ( i ) ⇔ ( ii ) ⇔ ( iii ) Theorem (Anantharaman-Delaroche, 2002) For G locally compact, we have ( i ) ⇒ ( ii ) ⇒ ( iii ) Tobe Deprez Class S for locally compact groups 15 / 27
Topological amenability Exactness and topological amenability Definition A group G is called exact if the operation of taking the reduced crossed product preserves short exact sequences. Consider the following conditions (i) G is exact, (ii) G � β lu G is amenable, (iii) C ∗ r ( G ) is exact (i.e. taking minimal tensor product preserves exactness) Definition Left-equivariant Stone-Čech compactification β lu G G -equiv f C ( β lu G ) ∼ = C lu b ( G ) G K � � λ g f − f � ∞ → 0 if g → e = � f ∈ C b ( G ) � � i ∃ ! G -equiv β f β lu G Tobe Deprez Class S for locally compact groups 16 / 27
Topological amenability Exactness and topological amenability Consider the following conditions (i) G is exact, (ii) G � β lu G is amenable, (iii) C ∗ r ( G ) is exact (i.e. taking minimal tensor product preserves exactness) Theorem (Kirchberg-Wasserman, 1999; Ozawa, 2000) For G countable, we have ( i ) ⇔ ( ii ) ⇔ ( iii ) Theorem (Anantharaman-Delaroche, 2002; Brodzki-Cave-Li, 2017) For G locally compact, we have ( i ) ⇔ ( ii ) ⇒ ( iii ) . ◮ Remark : for locally compact ( iii ) ⇒ ( i ) is open. Tobe Deprez Class S for locally compact groups 17 / 27
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