Strictly outer actions of groups and quantum groups

2002/11/30 by Stefaan Vaes · 1 citation
Mathematics · #math.OA #math.QA

paper · pdf

published as Journal für die reine und angewandte Mathematik 578 (2005), 147--184. · Completely rewritten and extended version. In particular, we discuss in detail strictly outer actions of compact and discrete quantum groups on injective factors

arxiv created 2003/05/23 · arxiv updated 2009/11/30

Abstract

An action of a locally compact group or quantum group on a factor is said to be strictly outer when the relative commutant of the factor in the crossed product is trivial. We show that all locally compact quantum groups can act strictly outerly on a free Araki-Woods factor and that all locally compact groups can act strictly outerly on the hyperfinite II₁ factor. We define a kind of Connes' T invariant for locally compact quantum groups and prove a link with the possibility of acting strictly outerly on a factor with a given T invariant. Necessary and sufficient conditions for the existence of strictly outer actions of compact Kac algebras on the hyperfinite II₁ factor are given.

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• Amenability of locally compact quantum groups via the long-time behavior of convolution semigroups

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