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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 II1 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 II1 factors

Let M be a II1 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 αi → α if and only if ||αi(x) − α(x)||2 → 0 for all x ∈ M. Theorem (Connes 1974) Let M be a II1 factor. The following are equivalent: Inn(M) is closed in Aut(M). For every bounded sequence (xn)n∈N in M such that ||xna − axn||2 → 0 for all a ∈ M, we have ||xn − τ(xn)||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 II1 factor. The hyperfinite II1 factor R =

n∈N M2(C) is not full. Indeed,

there is a non-trivial sequence of unitaries given by un = 1⊗n ⊗ 1 −1

• ⊗ 1 ⊗ · · ·

Theorem (Murray & von Neumann 1943) The free group factor L(F2) is full. In fact, they obtained the following inequality: ∀x ∈ L(F2), ||x − τ(x)||2 ≤ 14 max(||xa − ax||2, ||xb − bx||2) This is called the ”14ǫ-lemma”. It is a spectral gap property!

Amine Marrakchi Spectral gap characterization of full type III factors

Spectral gap characterization of full II1 factors

In 1976, Connes published his celebrated work on the classification

• f injective factors. As a key step in his proof of injectivity ⇒

hyperfiniteness, he proved the following remakable characterization

• f full factors:

Theorem (Connes 1976) Let M be a full II1 factor. There exist a family a1, . . . , an ∈ M and a constant C > 0 such that ∀x ∈ M, ||x − τ(x)||2 ≤ C max

1≤k≤n ||xak − akx||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 L2(M) associated to M. Every element a ∈ M acts on L2(M) on the left ξ → aξ and on the right ξ → ξa. We equip Aut(M) with the topology of pointwise convergence on L2(M), i.e αi → α if and only if ||αi(ξ) − α(ξ)||2 → 0 for all ξ ∈ L2(M). Theorem (Connes 1974) Let M be a factor. The following are equivalent: Inn(M) is closed in Aut(M). For every bounded sequence (xn)n∈N in M such that ||xnξ − ξxn||2 → 0 for all ξ ∈ L2(M), there exists λn ∈ C such that xn − λn → 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 ξ1, . . . , ξn ∈ L2(M) and a constant C > 0 such that ∀x ∈ M, ||x − ϕ(x)||ϕ ≤ C max

1≤k≤n ||xξk − ξkx||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 → σϕ

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 ut ∈ U(M) such that σψ

t = Ad(ut) ◦ σϕ t .

Two consequences: The one-parameter group δ : t ∈ R → [σϕ

t ] ∈ Out(M) does

not depend on ϕ. The crossed product c(M) = M ×σϕ 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 III1 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 III1 if ker δ = {0}. Proposition M is a factor of type III1 if and only if c(M) is a factor. Question: when is c(M) a full factor? Theorem (Shlyakhtenko 2004) Let M be a III1 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 III1 factor which admits an almost periodic state ⇒ δ(R) is compact.

Amine Marrakchi Spectral gap characterization of full type III factors

Type III1 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 ⋊σϕ R is full if and only if M ⋊σϕ Z is full. Theorem (Jones 1982) Let M be a full II1 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 III1 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 III1 factors: Corollary (M. 2016) Let M be a III1 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(L2(M)) and ρ : Mop → B(L2(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(Mop). Theorem (Connes 1976) Let M be a II1 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 II1 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