Explicit computations of all finite index bimodules for a family of - PowerPoint PPT Presentation

Explicit computations of all finite index bimodules for a family of II 1 factors Fields Institute Workshop Von Neumann Algebras Stefaan Vaes 1/20

II 1 factor without finite index subfactors Trivial subfactor : 1 ⊗ N ⊂ M n ( C ) ⊗ N . Take Γ = SL ( 2 , Q ) ⋉ Q 2 , with Γ ↷ ( X , µ) = ( X 0 , µ 0 ) Q 2 . Theorem (V, 2007) The II 1 factor M = L ∞ ( X ) ⋊ Ω α Γ has no non-trivial finite index subfactors if ◮ ( X 0 , µ 0 ) atomic with atoms of different weights, ◮ α ∈ R \ { 0 } and Ω α ∈ Z 2 ( Γ , S 1 ) defined by �� x 1 �� � � y 1 = exp ( 2 π i α( x 1 y 2 − x 2 y 1 )) Ω α , x 2 y 2 on Q 2 and extended to Γ by SL ( 2 , Q ) -invariance. Moreover, the II 1 factor M remembers ( X 0 , µ 0 ) and α . 2/20

Other results and plan of talk ◮ Introduce finite index bimodules and fusion algebra of II 1 factors. ◮ Present the first explicit computations of fusion algebras of II 1 factors : • Identification with Hecke algebras. • Crucial ingredients : Popa’s deformation/rigidity and cocycle superrigidity. ◮ Theorem (V, 2007, generalizing Popa – V, 2006). Every countable group arises as the outer automorphism group of a II 1 factor. 3/20

Connes’ correspondences A representation theory of II 1 factors Let M be a type II 1 factor with trace τ . ◮ A right M -module is a Hilbert space with a right action of M . Example : L 2 ( M , τ) M . � i ∈ I p i L 2 ( M ) ◮ Always, H M ≅ � and one defines dim ( H M ) = i τ( p i ) ∈ [ 0 , +∞ ] . Complete invariant of right M -modules. Definition An M - M -bimodule of finite Jones index, is an M - M -bimodule M H M dim ( H M ) < ∞ dim ( M H ) < ∞ . satisfying and Denote FAlg ( M ) the set of (equiv. classes) of finite index bimodules. 4/20

The structure of the fusion algebra FAlg( M ) Example : let α ∈ Aut ( M ) and define the M - M -bimodule H (α) on L 2 ( M ) by a · ξ · b = a ξα( b ) Precisely those M H M with dim ( M H ) = 1 = dim ( H M ) . Example : let M ⊂ M 1 be a finite index subfactor. Then, M L 2 ( M 1 ) M belongs to FAlg ( M ) . In general, FAlg( M ) carries the following structure. ◮ Direct sum of bimodules. ◮ Identity element : M L 2 ( M ) M . ◮ Connes’ tensor product H ⊗ M K . Example : H (α) ⊗ M H (β) ≅ H (α ◦ β) . ◮ Contragredient M H M of M H M . FAlg ( M ) consists of the generalized symmetries of M . 5/20

Group-like elements in FAlg( M ) We call M H M group-like if H ⊗ M H is the identity. Example of group-like element M H M : ◮ H = L 2 ( M ) p . ◮ α : M → pMp an isomorphism. ◮ a · ξ · b = a ξα( b ) . Set Out ( M ) = Aut ( M ) Inn ( M ) . Observation We have a short exact sequence e → Out ( M ) → { group-like bimodules } → F ( M ) → e where F ( M ) is the fundamental group of M . 6/20

Abstract fusion algebras Definition A fusion algebra A is a free N -module N [ G ] , equipped with � ◮ an associat. distribut. product : x ∗ y = mult ( z , x ∗ y ) z , z ∈G ◮ a multiplicative neutral element e ∈ G , ◮ a contragredient map x ֏ x which is ..., such that Frobenius reciprocity holds : for all x , y , z ∈ G , we have mult ( z , x ∗ y ) = mult ( x , z ∗ y ) = mult ( y , x ∗ z ) Examples ◮ N [ Γ ] for a group Γ . ◮ Rep ( G ) , the finite dim. unitary rep. of a compact group G . ◮ FAlg ( M ) of a II 1 factor M . 7/20

Fusion algebra of a Hecke pair [ Γ : g Γ g − 1 ∩ Γ ] < ∞ for all g ∈ G . Let Γ < G be a Hecke pair, i.e. Hecke fusion algebra H ( Γ < G ) = { ξ : Γ \ G / Γ → N | ξ has finite support } � ξ( h ) η( h − 1 g ) (ξ ∗ η)( g ) = h ∈ G / Γ Γ \ G / Γ is the set of irreducibles with fusion rules ... Example Let T be k -valent tree and G < Aut ( T ) countable dense subgroup. Choose a vertex e and set Γ = Stab e . Identify Γ \ G / Γ = Γ \ T ≅ N via the distance to e . Then, mult ( n , a ∗ b ) = # { t ∈ T | | et | = a , | ts | = b } when | es | = n . A way to understand the Hecke fusion algebra of PSL ( 2 , Z ) < PSL ( 2 , Q ) . 8/20

Computations of FAlg( M ) Data : action of Γ on a countable set I , base probability space ( X 0 , µ 0 ) . Action : Γ ↷ ( X , µ) = ( X 0 , µ 0 ) I . II 1 factor : M = L ∞ ( X ) ⋊ Γ . Theorem (V, 2007) Under the right conditions on Γ ↷ I and for ( X 0 , µ 0 ) atomic with distinct weights, we have � → � → H ( Γ < G ) Rep fin ( Γ ) FAlg ( M ) where G = Comm Perm I ( Γ ) = commensurator of Γ inside Perm I . For every Hecke pair Γ < G , one defines H rep ( Γ < G ) . Then, FAlg ( M ) ≅ H rep ( Γ < G ) . 9/20

Concrete examples Recall : M = L ∞ ( X ) ⋊ Γ Γ ↷ ( X , µ) = ( X 0 , µ 0 ) I . and FAlg ( M ) Γ ↷ I SL n ( Z ) ⋉ Q n Q n for n ≥ 2. H rep ( SL n ( Z ) < GL n ( Q )) ↷ H rep ( Λ < Comm PGL n ( Q ) ( Λ )) . Λ < PSL n ( Q ) proper subgroup, relative ICC, Γ = Λ × PSL n ( Q ) ↷ PSL n ( Q ) by left-right action. H rep ( R ∗ ⋉ R < Q ∗ ⋉ Q ) Z < R < Q strict inclusions of rings, i.e. R = Z [ P − 1 ] Relation with Bost- Write some Λ 0 < Λ = SL 2 ( Q )⋉ Q 2 . Connes Hecke algebra. Take Λ 0 × Λ ↷ Λ . Add 2-cocycle. SL 2 ( Q ) ⋉ Q 2 ↷ Q 2 Trivial. Add 2-cocycle. 10/20

M = L ∞ ( X ) ⋊ Γ Γ ↷ ( X , µ) = ( X 0 , µ 0 ) I . We continue with and FAlg ( M ) ≅ H rep ( Γ < Comm Perm I ( Γ )) Our isomorphism works for Good actions of good groups A condition on the group Γ : ◮ Γ admits an infinite, almost normal subgroup with relative (T). Some conditions on the action Γ ↷ I : ◮ Transitivity. ◮ Stab i 0 acts with infinite orbits on I − { i 0 } . ◮ Minimal condition on stabilizers : no infinite sequence ( i n ) with Stab ( i 0 , . . . , i n ) strictly decreasing. ◮ A faithfulness condition of Γ → Perm I . About the minimal condition on stabilizers. Automatic if Γ ↷ I embeds in GL ( V ) ⋉ V ↷ V for fin.dim. V . For left-right action Λ × Λ ↷ Λ : equivalent with minimal condition on centralizers of Λ . 11/20

How to find all finite index M - M - bimodules Take Γ ↷ I a good action of a good group. M = L ∞ ( X ) ⋊ Γ Γ ↷ ( X , µ) = ( X 0 , µ 0 ) I . Let and Take a finite index M - M -bimodule M H M . 1 The bimodule H contains a finite index L ( Γ ) - L ( Γ ) -subbimodule. Main ingredients : Popa’s deformation/rigidity and the minimal condition on stabilizers. 2 The bimodule H contains a finite index L ∞ ( X ) - L ∞ ( X ) -subbimodule. 3 The bimodule H belongs to H rep ( Γ < Comm Aut ( X ,µ) ( Γ )) Main ingredient : Popa’s cocycle superrigidity. 4 Identification of Comm Aut ( X ,µ) ( Γ ) and Comm Perm I ( Γ ) Main ingredient : Stab i 0 acts with infinite orbits on I − { i 0 } . Attention : two difficult slides follow with steps 1 and 2. 12/20

Every bimodule is L ( Γ ) -preserving (Step 1) M = L ∞ ( X ) ⋊ Γ Γ ↷ ( X , µ) = ( X 0 , µ 0 ) I . Let and Take a finite index M - M -bimodule M H M . Simplifying assumptions : H = H (ψ) for ψ ∈ Aut ( M ) and Γ has property (T). Aim : ψ( L ( Γ )) and L ( Γ ) are unitarily conjugate. ◮ Popa’s malleability (roughly) : • flow (α t ) t ∈ R on L ∞ ( X × X ) , • commuting with diagonal Γ -action, • connecting id = α 0 with flip = α 1 . ◮ Extend α t to L ∞ ( X × X ) ⋊ Γ . Set P = ψ( L ( Γ )) ⊂ ( L ∞ ( X ) ⊗ 1 ) ⋊ Γ . By property (T), α 0 ( P ) and α 1 ( P ) are unitarily conjugate. Then, ψ( L ( Γ )) and L ( Γ ) are unitarily conjugate. ◮ Popa proves this for Γ ↷ ( X , µ) mixing (meaning here that Stab i is finite for all i ∈ I ) Mixing is replaced by minimal condition on stabilizers. 13/20

Every L ( Γ ) -preserving bimodule is Cartan-preserving (Step 2) M = L ∞ ( X ) ⋊ Γ Γ ↷ ( X , µ) = ( X 0 , µ 0 ) I . Let and Take a finite index M - M -bimodule M H M . Simplifying assumptions : H = H (ψ) for ψ ∈ Aut ( M ) and ψ( L ( Γ )) = L ( Γ ) (because of Step 1). Aim : ψ( L ∞ ( X )) and L ∞ ( X ) are unitarily conjugate. ◮ Fix i 0 ∈ I and set Γ 0 = Stab i 0 . Recall : Stab i 0 acts with infinite orbits on I − { i 0 } . First aim : ψ( L ( Γ 0 )) and L ( Γ 0 ) are unitarily conjugate. ◮ If not, ψ( L ( Γ 0 )) will be far from L ( Γ 0 ) leading to ψ( L ( Γ 0 )) ′ ∩ M ⊂ L ( Γ ) . Contradiction. ◮ So, we may assume ψ( L ( Γ 0 )) = L ( Γ 0 ) . Take relative commutants : � L ∞ � ( X 0 , µ 0 ) { i 0 } � � = L ∞ � ( X 0 , µ 0 ) { i 0 } � ψ ⋊ Γ 0 ⋊ Γ 0 . ◮ Play a game to get ψ( L ∞ ( X )) and L ∞ ( X ) conjugate. 14/20

Cartan preserving bimodules and cocycle superrigidity M = L ∞ ( X ) ⋊ Γ Γ ↷ ( X , µ) = ( X 0 , µ 0 ) I . Let and Take a finite index M - M -bimodule M H M . Simplifying assumptions : H = H (ψ) for ψ ∈ Aut ( M ) and ψ( L ∞ ( X )) = L ∞ ( X ) (because of Step 2). Restriction of ψ to L ∞ ( X ) yields an orbit equivalence ∆ : X → X : ∆ ( Γ · x ) = Γ · ∆ ( x ) a.e. Associated Zimmer 1-cocycle : ∆ ( g · x ) = ω( g , x ) · ∆ ( x ) . Cocycle superrigidity theorem (Popa, 2005) Let Γ ↷ ( X , µ) = ( X 0 , µ 0 ) I . Assume ◮ H < Γ almost normal and relative (T), ◮ H acts with infinite orbits on I , ◮ G either a countable or a compact group. Every 1-cocycle for Γ ↷ ( X , µ) with values in G , is cohomologous with a homomorphism Γ → G . 15/20