# 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}
}
• Published 27 October 2015
• Mathematics
• Communications in Mathematical Physics
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…
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## References

SHOWING 1-10 OF 53 REFERENCES

• N. Ozawa
• Mathematics
Compositio 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
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
• 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
• 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
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
• Mathematics
Compositio 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
• 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
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