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A remark on fullness of some group measure space von Neumann algebras

Part of: Abstract harmonic analysis Selfadjoint operator algebras

Published online by Cambridge University Press:  02 November 2016

Narutaka Ozawa
Narutaka Ozawa*
Affiliation:
RIMS, Kyoto University, 606-8502 Kyoto, Japan email [email protected]
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Abstract

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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 $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ on a standard measure space, the group measure space von Neumann algebra $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(X)$ is full. In this paper, we prove the same property for a wider class of groups, notably including $\text{SL}(3,\mathbb{Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $\unicode[STIX]{x1D6E4}\leqslant G$, and any closed non-amenable subgroup $H\leqslant G$, the non-singular action $\unicode[STIX]{x1D6E4}\curvearrowright G/H$ is strongly ergodic and the von Neumann factor $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(G/H)$ is full.

Keywords

full factorsstrongly ergodic actions

MSC classification

Primary: 46L36: Classification of factors
Secondary: 46L10: General theory of von Neumann algebras 43A07: Means on groups, semigroups, etc.; amenable groups
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 12 , December 2016 , pp. 2493 - 2502
Copyright
© The Author 2016 

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