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Fullness of crossed products of factors by discrete groups

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  23 April 2019

Amine Marrakchi
Amine Marrakchi*
Affiliation:
Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, Orsay91405, France (amine.marrakchi@math.u-psud.fr)
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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 $. When Γ is amenable, we obtain a complete characterization: the crossed product factor $M \rtimes _\sigma \Gamma $ is full if and only if M is full and the quotient map $\overline {\sigma } : \Gamma \rightarrow {\rm out}(M)$ has finite kernel and discrete image. This answers the question of Jones from [11]. When M is full and Γ is arbitrary, we give a sufficient condition for $M \rtimes _\sigma \Gamma $ to be full which generalizes both Jones' criterion and Choda's criterion. In particular, we show that if M is any full factor (possibly of type III) and Γ is a non-inner amenable group, then the crossed product $M \rtimes _\sigma \Gamma $ is full.

Keywords

Full factorscrossed productspectral gapAmenable grouptype III factors

MSC classification

Primary: 46L10: General theory of von Neumann algebras
Secondary: 46L36: Classification of factors 46L40: Automorphisms 46L55: Noncommutative dynamical systems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2368 - 2378
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Copyright
Copyright © Royal Society of Edinburgh 2019

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