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Communications in Mathematical Physics

, Volume 350, Issue 3, pp 897–916 | Cite as

Solidity of Type III Bernoulli Crossed Products

  • Amine MarrakchiEmail author
Article

Abstract

We generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra A 0, any faithful normal state \({\varphi_0}\) and any discrete group \({\Gamma}\), the associated Bernoulli crossed product von Neumann algebra \({M=(A_0,\varphi_0)^{\overline{\otimes} \Gamma} \rtimes \Gamma}\) is solid relatively to \({\mathcal{L}(\Gamma)}\). In particular, if \({\mathcal{L}(\Gamma)}\) is solid then M is solid and if \({\Gamma}\) is non-amenable and \({A_0 \neq \mathbb{C}}\) then M is a full prime factor. This gives many new examples of solid or prime type III factors. Following Chifan and Ioana, we also obtain the first examples of solid non-amenable type III equivalence relations.

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.École Normale SupérieureParis Cedex 05France
  2. 2.Laboratoire de Mathématiques d’OrsayUniversité Paris-Sud, Université Paris-SaclayOrsayFrance

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