Spectral gap characterization of full type III factors Amine Marrakchi Advisor: Cyril Houdayer Ecole Normale Sup´ erieure Universit´ e Paris-Sud July 25, 2016 Amine Marrakchi Spectral gap characterization of full type III factors
Introduction Our goal in this talk will be to: Define full factors, with motivations and examples. explain the spectral gap characterization of full II 1 factors by Connes. give a type III analog of Connes’s result. discuss applications and open questions. Amine Marrakchi Spectral gap characterization of full type III factors
Full II 1 factors Let M be a II 1 factor with its unique trace τ : M → C , τ (1) = 1 and τ ( ab ) = τ ( ba ) for all a , b ∈ M . Let || x || 2 := τ ( x ∗ x ) 1 / 2 for all x ∈ M . We equip the automorphism group Aut ( M ) with the topology of pointwise || · || 2 -convergence, i.e α i → α if and only if || α i ( x ) − α ( x ) || 2 → 0 for all x ∈ M . Theorem (Connes 1974) Let M be a II 1 factor. The following are equivalent: Inn ( M ) is closed in Aut ( M ) . For every bounded sequence ( x n ) n ∈ N in M such that || x n a − ax n || 2 → 0 for all a ∈ M, we have || x n − τ ( x n ) || 2 → 0 . When these conditions are satisfied we say that M is full. Amine Marrakchi Spectral gap characterization of full type III factors
Examples Fullness was already studied by Murray & von Neumann in order to obtain an example of a non-hyperfinite II 1 factor. The hyperfinite II 1 factor R = � n ∈ N M 2 ( C ) is not full. Indeed, there is a non-trivial sequence of unitaries given by � 1 � 0 u n = 1 ⊗ n ⊗ ⊗ 1 ⊗ · · · 0 − 1 Theorem (Murray & von Neumann 1943) The free group factor L ( F 2 ) is full. In fact, they obtained the following inequality: ∀ x ∈ L ( F 2 ) , || x − τ ( x ) || 2 ≤ 14 max( || xa − ax || 2 , || xb − bx || 2 ) This is called the ”14 ǫ -lemma”. It is a spectral gap property! Amine Marrakchi Spectral gap characterization of full type III factors
Spectral gap characterization of full II 1 factors In 1976, Connes published his celebrated work on the classification of injective factors. As a key step in his proof of injectivity ⇒ hyperfiniteness, he proved the following remakable characterization of full factors: Theorem (Connes 1976) Let M be a full II 1 factor. There exist a family a 1 , . . . , a n ∈ M and a constant C > 0 such that ∀ x ∈ M , || x − τ ( x ) || 2 ≤ C max 1 ≤ k ≤ n || xa k − a k x || 2 There are two main steps in the proof: Use ultraproducts and singular states to produce ”microscopic” almost central projections. Via a maximality argument, patch this ”microscopic” projections to produce a ”macroscopic” projection. Amine Marrakchi Spectral gap characterization of full type III factors
Full type III factors Let M be a von Neumann algebra. There is a canonical Hilbert space L 2 ( M ) associated to M . Every element a ∈ M acts on L 2 ( M ) on the left ξ �→ a ξ and on the right ξ �→ ξ a . We equip Aut ( M ) with the topology of pointwise convergence on L 2 ( M ), i.e α i → α if and only if || α i ( ξ ) − α ( ξ ) || 2 → 0 for all ξ ∈ L 2 ( M ). Theorem (Connes 1974) Let M be a factor. The following are equivalent: Inn ( M ) is closed in Aut ( M ) . For every bounded sequence ( x n ) n ∈ N in M such that || x n ξ − ξ x n || 2 → 0 for all ξ ∈ L 2 ( M ) , there exists λ n ∈ C such that x n − λ n → 0 strongly. If these conditions are satified we say that M is full. Amine Marrakchi Spectral gap characterization of full type III factors
Spectral gap characterization of full type III factors Let M be a type III factor, i.e a factor which does not have a trace. For ϕ a faithful normal state, we let || x || ϕ := ϕ ( x ∗ x ) 1 / 2 for all x ∈ M . Theorem (M. 2016) Let M be a full type III factor. There exist a faithful normal state ϕ , a family ξ 1 , . . . , ξ n ∈ L 2 ( M ) and a constant C > 0 such that ∀ x ∈ M , || x − ϕ ( x ) || ϕ ≤ C max 1 ≤ k ≤ n || x ξ k − ξ k x || 2 The proof is inspired from Connes’s original proof but the first step is very different. It relies on a new tool: the Groh-Raynaud ultraproduct, and the recent work of Ando and Haagerup. Amine Marrakchi Spectral gap characterization of full type III factors
Modular theory Let M be a von Neumann algebra with a faithful normal state ϕ . Tomita-Takesaki’s theory: the state ϕ produces a one-parameter group of automorphisms t ∈ R �→ σ ϕ t ∈ Aut ( M ) called the modular flow of ϕ . This modular flow is trivial iff ϕ is a trace. Connes’s cocycle theorem: if ψ is an other state, there exists a cocyle u t ∈ U ( M ) such that σ ψ t = Ad ( u t ) ◦ σ ϕ t . Two consequences: The one-parameter group δ : t ∈ R �→ [ σ ϕ t ] ∈ Out ( M ) does not depend on ϕ . The crossed product c ( M ) = M × σ ϕ R does not depend on ϕ up to canonical isomorphism. It is called the core of M . Amine Marrakchi Spectral gap characterization of full type III factors
Type III 1 factors with full core Let M be a factor of type III ( ⇔ δ is non-trivial). Then ker δ may be any countable subgroup of R . We say that M is of type III 1 if ker δ = { 0 } . Proposition M is a factor of type III 1 if and only if c ( M ) is a factor. Question : when is c ( M ) a full factor? Theorem (Shlyakhtenko 2004) Let M be a III 1 factor. If c ( M ) is full then M is full and δ : R → Out ( M ) is a homeomorphism on its range. The condition on δ may fail, for example, if M is a full III 1 factor which admits an almost periodic state ⇒ δ ( R ) is compact. Amine Marrakchi Spectral gap characterization of full type III factors
Type III 1 factors with full core Theorem (Ueda-Tomatsu 2014) The converse is true for free product factors M ∗ N and Bernoulli crossed products ( � G M ) ⋊ G. A key lemma: δ : R → Out ( M ) is a homeomorphism on its range if and only if δ ( Z ) is discrete. c ( M ) = M ⋊ σ ϕ R is full if and only if M ⋊ σ ϕ Z is full. Theorem (Jones 1982) Let M be a full II 1 factor and G a countable subgroup of Aut ( M ) . If the image of G in Out ( M ) = Aut ( M ) / Inn ( M ) is discrete, then the crossed product M ⋊ G is also a full factor. Amine Marrakchi Spectral gap characterization of full type III factors
Type III 1 factors with full core Using the spectral gap characterization of full type III factors, one can adapt Jones’s proof to the type III case: Theorem (M. 2016) Let M be a full type III factor and G a countable subgroup of Aut ( M ) . If the image of G in Out ( M ) = Aut ( M ) / Inn ( M ) is discrete, then the crossed product M ⋊ G is also a full factor. Using this, we deduce that the conjecture of Ueda and Tomatsu holds for all III 1 factors: Corollary (M. 2016) Let M be a III 1 factor. Then c ( M ) is full if and only if M is full and δ : R → Out ( M ) is a homeomorphism on its range. Amine Marrakchi Spectral gap characterization of full type III factors
Open questions Let λ : M → B ( L 2 ( M )) and ρ : M op → B ( L 2 ( M )) be the left and right regular representations of M . We define C ∗ λ,ρ ( M ) the C ∗ -algebra generated by λ ( M ) and ρ ( M op ). Theorem (Connes 1976) Let M be a II 1 factor. Then M is full if and only if C ∗ λ,ρ ( M ) contains the compact operators. Corollary (Connes 1976) Let M and N be two full II 1 factor. Then M ⊗ N is also full. Let M be a full type III factor. Is it true that C ∗ λ,ρ ( M ) contains the compact operators? Let M and N be two full type III factors. Is it true that M ⊗ N is also full? Amine Marrakchi Spectral gap characterization of full type III factors
Thank you for your attention! Amine Marrakchi Spectral gap characterization of full type III factors
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