# Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence

@article{Houdayer2015BiExactGS, title={Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence}, author={Cyril Houdayer and Yusuke Isono}, journal={Communications in Mathematical Physics}, year={2015}, volume={348}, pages={991-1015} }

AbstractWe investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras $${M = B \rtimes \Gamma}$$M=B⋊Γ arising from arbitrary actions $${\Gamma \curvearrowright B}$$Γ↷B of bi-exact discrete groups (e.g. free groups) on amenable von Neumann algebras. We prove a spectral gap rigidity result for the central sequence algebra $${N' \cap M^\omega}$$N′∩Mω of any nonamenable von Neumann subalgebra with normal expectation $${N \subset M}$$N⊂M. We use this result to…

## 19 Citations

### A remark on fullness of some group measure space von Neumann algebras

- MathematicsCompositio Mathematica
- 2016

Recently Houdayer and Isono have proved, among other things, that every biexact group $\unicode[STIX]{x1D6E4}$ has the property that for any non-singular strongly ergodic essentially free action…

### Strongly ergodic equivalence relations: spectral gap and type III invariants

- MathematicsErgodic Theory and Dynamical Systems
- 2017

We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product…

### Fullness of crossed products of factors by discrete groups

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2020

Abstract Let M be an arbitrary factor and $\sigma : \Gamma \curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes _\sigma \Gamma $.…

### Solidity of Type III Bernoulli Crossed Products

- Mathematics
- 2016

AbstractWe generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra A0, any faithful normal state $${\varphi_0}$$φ0 and any discrete group…

### Boundary and Rigidity of Nonsingular Bernoulli Actions

- MathematicsCommunications in Mathematical Physics
- 2021

Let G be a countable discrete group and consider a nonsingular Bernoulli shift action G↷∏g∈G({0,1},μg)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}…

### CAT(0) cube complexes and inner amenability

- Mathematics
- 2019

We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group $G$ on a finite dimensional irreducible CAT(0) cube…

### Nonsingular Bernoulli actions of arbitrary Krieger type

- MathematicsAnalysis & PDE
- 2022

We prove that every infinite amenable group admits Bernoulli actions of any possible Krieger type, including type $II_\infty$ and type $III_0$. We obtain this result as a consequence of general…

### Solidity of Type III Bernoulli Crossed Products

- Materials ScienceCommunications in Mathematical Physics
- 2016

We generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra A0, any faithful normal state φ0\documentclass[12pt]{minimal} \usepackage{amsmath}…

### Properly Proximal von Neumann Algebras

- Mathematics
- 2022

. We introduce the notion of proper proximality for ﬁnite von Neumann algebras, which naturally extends the notion of proper proximality for groups. Apart from the group von Neumann algebras of…

### Co-spectral radius, equivalence relations and the growth of unimodular random rooted trees

- Mathematics
- 2022

. We deﬁne the co-spectral radius of inclusions S ≤ R of discrete, probability measure-preserving equivalence relations, as the sampling exponent of a generating random walk on the ambient relation.…

## References

SHOWING 1-10 OF 53 REFERENCES

### A remark on fullness of some group measure space von Neumann algebras

- MathematicsCompositio Mathematica
- 2016

Recently Houdayer and Isono have proved, among other things, that every biexact group $\unicode[STIX]{x1D6E4}$ has the property that for any non-singular strongly ergodic essentially free action…

### Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I

- Mathematics
- 2006

We consider crossed product II1 factors $M = N\rtimes_{\sigma}G$, with G discrete ICC groups that contain infinite normal subgroups with the relative property (T) and σ trace preserving actions of G…

### Local spectral gap in simple Lie groups and applications

- Mathematics
- 2015

We introduce a novel notion of local spectral gap for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action $$\Gamma \curvearrowright…

### Asymptotic structure of free Araki–Woods factors

- Mathematics
- 2014

The purpose of this paper is to investigate the structure of Shlyakhtenko’s free Araki–Woods factors using the framework of ultraproduct von Neumann algebras. We first prove that all the free…

### Cartan subalgebras of amalgamated free product II$_1$ factors

- Mathematics
- 2012

We study Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtain several uniqueness and non-existence results. We prove that if $\Gamma$ belongs to a large class of…

### Rigidity of free product von Neumann algebras

- MathematicsCompositio Mathematica
- 2016

Let $I$ be any nonempty set and let $(M_{i},\unicode[STIX]{x1D711}_{i})_{i\in I}$ be any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class…

### On a tensor product $C^{*}$-algebra associated with the free group on two generators

- Mathematics
- 1975

Let $G$ be the free group on two generators, and $L^{2}$ the Hilbert space of square summable complex valued functions on $G$ . Let $\mathcal{L}$ and $\mathcal{R}$ be the $C^{*}-$ algebras generated…

### C*-Algebras and Finite-Dimensional Approximations

- Mathematics
- 2008

Fundamental facts Basic theory: Nuclear and exact $\textrm{C}^*$-algebras: Definitions, basic facts and examples Tensor products Constructions Exact groups and related topics Amenable traces and…