II 1 factors with a unique Cartan decomposition BIRS, Banff, June - PowerPoint PPT Presentation

II 1 factors with a unique Cartan decomposition BIRS, Banff, June 2012 Stefaan Vaes ∗ ∗ Supported by ERC Starting Grant VNALG-200749 1/20

The group measure space construction Input : a countable group Γ and a probability measure preserving action Γ � ( X , µ ). Consider Γ � L ∞ ( X ) by ( g · F )( x ) = F ( g − 1 · x ). Output : the von Neumann algebra M = L ∞ ( X ) ⋊ Γ. ◮ M is generated by a copy of L ∞ ( X ) and unitaries ( u g ) g ∈ Γ . ◮ We have u g u h = u gh and u g Fu ∗ g = g · F . Think of the semi-direct product of groups. ◮ M has a trace τ : M → C given by � τ ( Fu g ) = 0 if g � = e and τ ( F ) = F d µ . Main research theme Classify crossed products L ∞ ( X ) ⋊ Γ in terms of the group action data ! 2/20

Freeness, ergodicity, II 1 factors Fix a probability measure preserving (pmp) action Γ � ( X , µ ). ◮ The subalgebra L ∞ ( X ) ⊂ L ∞ ( X ) ⋊ Γ is maximal abelian if and only if Γ � X is free : for all g � = e and for a.e. x ∈ X , we have g · x � = x . ◮ Assuming that Γ � X is free, we get that L ∞ ( X ) ⋊ Γ is a factor if and only if Γ � X is ergodic : all Γ-invariant subsets of X have measure 0 or 1. We only consider free ergodic pmp actions Γ � ( X , µ ). Then, L ∞ ( X ) ⋊ Γ is a II 1 factor. 3/20

Examples of free ergodic pmp actions ◮ Irrational rotation Z � T given by n · z = exp(2 π i α n ) z for a fixed irrational number α ∈ R \ Q . ◮ Bernoulli action Γ � ( X 0 , µ 0 ) Γ given by ( g · x ) h = x hg . ◮ The action SL( n , Z ) � T n = R n / Z n . ◮ The action Γ � G / Λ for lattices Γ , Λ < G . ◮ Certain profinite actions Γ � lim − Γ / Γ n with [Γ : Γ n ] < ∞ . ← They give all rise to II 1 factors L ∞ ( X ) ⋊ Γ. 4/20

Cartan subalgebras Definition A Cartan subalgebra A ⊂ M is a maximal abelian subalgebra whose normalizer N M ( A ) = { u ∈ U ( M ) | uAu ∗ = A } generates M . Examples : ◮ L ∞ ( X ) ⊂ L ∞ ( X ) ⋊ Γ if Γ � ( X , µ ) is a free ergodic pmp action. ◮ L ∞ ( X ) ⊂ L( R ) if R is a countably infinite ergodic pmp equivalence relation on ( X , µ ). The previous example corresponds to the orbit equivalence relation. ◮ Generic case : L ∞ ( X ) ⊂ L Ω ( R ) with a scalar 2-cocycle Ω. Conclusion Uniqueness of Cartan subalgebras = reducing the classification of L ∞ ( X ) ⋊ Γ to the classification of equivalence relations. 5/20

Amenable versus nonamenable Connes’ theorem (1975) All amenable II 1 factors are isomorphic with the unique hyperfinite II 1 factor R . ◮ LΓ ∼ = R for all amenable icc groups Γ. ◮ L ∞ ( X ) ⋊ Γ ∼ = R for all free ergodic pmp actions of an infinite amenable group Γ. Connes-Feldman-Weiss (1981) All amenable II 1 equivalence relations are isomorphic with the unique hyperfinite II 1 equivalence relation. The hyperfinite II 1 factor R has a unique Cartan subalgebra up to conjugacy by an automorphism: if A , B ⊂ R are Cartan, there exists α ∈ Aut( R ) with α ( A ) = B . 6/20

Non-uniqueness of Cartan subalgebras Connes-Jones, 1981 : II 1 factors with at least two Cartan subalgebras that are non-conjugate by an automorphism. Strongly ergodic actions Γ � ( X , µ ) such that M = L ∞ ( X ) ⋊ Γ is McDuff, i.e. M ∼ = M ⊗ R . Voiculescu, 1995 : L F n , 2 ≤ n ≤ ∞ , has no Cartan subalgebra. p ) ⋊ ( Z 2 ⋊ SL(2 , Z )) has Ozawa-Popa, 2007 : the II 1 factor M = L ∞ ( Z 2 two non-conjugate Cartan subalgebras, namely L ∞ ( Z 2 p ) and L( Z 2 ). Speelman-V, 2011 : II 1 factors M = L ∞ ( X ) ⋊ Γ with many concrete non-conjugate Cartan subalgebras, given by L ∞ ( X / H i ) ⊗ L H i for a family of abelian normal subgroups H i ⊳ Γ. ◮ Examples of II 1 factors M such that unitary conjugacy of Cartan subalgebras is non-smooth. ◮ Examples of II 1 factors M such that automorphic conjugacy of Cartan subalgebras is complete analytic. 7/20

Main result Theorem (Popa - V, 2011) If F n � ( X , µ ) is an arbitrary free ergodic pmp action, then L ∞ ( X ) is the unique Cartan subalgebra of L ∞ ( X ) ⋊ F n , up to unitary conjugacy. Consider wreath product groups H ≀ Γ = H (Γ) ⋊ Γ. Corollary, using Gaboriau’s work ◮ If H is an abelian group and n � = m , then we have L( H ≀ F n ) �∼ = L( H ≀ F m ) . Of course we need H � = { e } . ◮ If n � = m and F n � X , F m � Y are arbitrary free ergodic probability measure preserving actions, then L ∞ ( X ) ⋊ F n �∼ = L ∞ ( Y ) ⋊ F m . 8/20

The first II 1 factors with unique Cartan subalgebra From now on : unique Cartan = unique up to unitary conjugacy. Ozawa-Popa, 2007 If F n � ( X , µ ) is a profinite free ergodic pmp action, then L ∞ ( X ) is the unique Cartan subalgebra of L ∞ ( X ) ⋊ F n . Profinite ergodic action : Γ � lim − Γ / Γ n where [Γ : Γ n ] < ∞ . ← First ingredient : The complete metric approximation property of the free group, and of all its crossed products by profinite actions. Second ingredient : Popa’s malleable deformation of any crossed product L ∞ ( X ) ⋊ F n . We explain both ingredients, and their gradual improvements up to our main result, in the next slides. 9/20

Complete metric approximation property (CMAP) Herz-Schur multipliers and CMAP (Haagerup, 1978) ◮ We call f : Γ → C a Herz-Schur multiplier if the linear map LΓ → LΓ : u g �→ f ( g ) u g is completely bounded. We denote by � f � cb the cb-norm of that map. ◮ We say that Γ has CMAP if there exists a sequence of finitely supported Herz-Schur multipliers f n : Γ → C tending to 1 pointwise and satisfying lim sup n � f n � cb = 1. Examples : ◮ Z has CMAP by Fej´ er summation of Fourier series. ◮ Amenable groups have CMAP : the f n can be taken positive definite. ◮ F n has CMAP by suitably cutting off the maps g �→ ρ | g | where 0 < ρ < 1, ρ → 1. ◮ CMAP is stable under free products, direct products, and subgroups. 10/20

CMAP for II 1 factors and consequences Definition A tracial von Neumann algebra ( M , τ ) is said to have CMAP if there exist normal finite rank φ n : M → M such that lim n � φ n ( x ) − x � 2 = 0 for all x ∈ M and such that lim sup n � φ n � cb = 1. ◮ LΓ has CMAP if and only if Γ has CMAP. ◮ If Γ has CMAP and Γ � ( X , µ ) is profinite, L ∞ ( X ) ⋊ Γ has CMAP. A partial converse: theorem by Ozawa-Popa, 2007 If ( M , τ ) has CMAP and A ⊂ M is an abelian subalgebra, then the action N M ( A ) � A given by conjugation is almost profinite (is weakly compact). Conclusion : if M = L ∞ ( X ) ⋊ Γ with Γ CMAP and Γ � X profinite, and if A ⊂ M is another Cartan subalgebra, then N M ( A ) � A is almost a profinite action. 11/20

Popa’s malleable deformations Consider a crossed product M = L ∞ ( X ) ⋊ F n . ◮ For every 0 < ρ < 1, we have a unital completely positive map ψ ρ : M → M given by ψ ρ ( bu g ) = ρ | g | bu g for all b ∈ L ∞ ( X ), g ∈ F n . ◮ One can dilate the family ( ψ ρ ) into a malleable deformation : we have M ⊂ � M , together with a 1-parameter group of automorphisms α t ∈ Aut( � M ) such that ψ ρ t ( x ) = E M ( α t ( x )) for all x ∈ M . Ozawa-Popa (2007) : if A ⊂ M is a weakly compact Cartan subalgebra, the malleable deformation can be used to prove that A must be unitarily conjugate with L ∞ ( X ). Uniqueness of Cartan subalgebras for profinite crossed products L ∞ ( X ) ⋊ F n follows. 12/20

More groups with malleable deformations Free group F n : the function g �→ | g | is conditionally of negative type g �→ ρ | g | for a fixed 0 < ρ < 1 is positive definite, and proper. and tends to 1 pointwise if ρ → 1. Most general conditionally negative type function on a countable group Γ : functions of the form g �→ � c ( g ) � 2 , where c : Γ → H R is a 1-cocycle into the orthogonal representation π : Γ → O ( H R ), i.e. a map satisfying c ( gh ) = c ( g ) + π ( g ) c ( h ). Theorem (Ozawa-Popa, 2008) Assume that Γ is nonamenable, has CMAP and that • Γ admits a proper 1-cocycle into an orthogonal representation that is weakly contained in the regular representation, • Γ � ( X , µ ) is a profinite action, then L ∞ ( X ) is the unique Cartan subalgebra of L ∞ ( X ) ⋊ Γ. 13/20

Weak amenability Definition (Cowling-Haagerup, 1988) A group Γ is weakly amenable if there exists a sequence of finitely supported Herz-Schur multipliers f n → C tending to 1 pointwise such that lim sup n � f n � cb < ∞ . (The optimal value of this lim sup is the Cowling-Haagerup constant of Γ.) Ozawa, 2010 : in all the previous results, CMAP may be replaced by weak amenability. If Γ � X is a profinite action and M = L ∞ ( X ) ⋊ Γ, we still have that any Cartan subalgebra A ⊂ M is “weakly compact”. Importance of this improvement : all Gromov hyperbolic groups are weakly amenable (Ozawa, 2007). 14/20