Actions of dense subgroups of compact groups and \textrmII₁-factors with the Haagerup property

2005/10/14 by Paul Jolissaint, Jolissaint, Paul
Mathematics · #20H05 #28D05 #46L10 #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:20H05 #msc:28D05 #msc:46L10

paper · pdf · doi:10.48550/arxiv.math/0510301

15 pages

arxiv created 2005/10/14 · arxiv updated 2009/12/01

Abstract

Let M be a finite von Neumann algebra with the Haagerup property, and let G be a compact group that acts continuously on M and that preserves some finite trace τ. We prove that if Γ is a countable subgroup of G which has the Haagerup property, then the crossed product algebra M\rtimesΓ has also the Haagerup property. In particular, we study some ergodic, non-weakly mixing actions of groups with the Haagerup property on finite, injective von Neumann algebras, and we prove that the associated crossed products von Neumann algebras are \textrmII₁-factors with the Haagerup property. If moreover the actions have Property (τ), then the latter factors are full.

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