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Communications in Mathematical Physics

, Volume 348, Issue 3, pp 991–1015 | Cite as

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

  • Cyril HoudayerEmail author
  • Yusuke Isono
Article

Abstract

We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras \({M = B \rtimes \Gamma}\) arising from arbitrary actions \({\Gamma \curvearrowright B}\) 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 \({N' \cap M^\omega}\) of any nonamenable von Neumann subalgebra with normal expectation \({N \subset M}\). We use this result to show that for any strongly ergodic essentially free nonsingular action \({\Gamma \curvearrowright (X, \mu)}\) of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor \({L^\infty(X) \rtimes \Gamma}\) has no nontrivial central sequence. Using recent results of Boutonnet et al. (Local spectral gap in simple Lie groups and applications, 2015), we construct, for every \({0 < \lambda \leq 1}\), a type \({{\rm III_\lambda}}\) strongly ergodic essentially free nonsingular action \({\mathbf{F}_\infty \curvearrowright (X_\lambda, \mu_\lambda)}\) of the free group \({{\mathbf{F}}_\infty}\) on a standard probability space so that the corresponding group measure space type \({{\rm III_\lambda}}\) factor \({L^\infty(X_\lambda, \mu_\lambda) \rtimes \mathbf{F}_\infty}\) has no nontrivial central sequence by our main result. In particular, we obtain the first examples of group measure space type \({{\rm III}}\) factors with no nontrivial central sequence.

Keywords

Discrete Group Faithful State Ergodic Action Countable Discrete Group Group Measure Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’OrsayUniversité Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance
  2. 2.RIMSKyoto UniversityKyotoJapan

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