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


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.


Discrete Group Faithful State Ergodic Action Countable Discrete Group Group Measure Space 
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© 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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