Maximal subgroups and von Neumann subalgebras with the Haagerup property

Abstract

We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside ${\mathbb{Z}}^2⋊S{L}_2(\mathbb{Z})$ and obtain several explicit instances where maximal Haagerup subgroups yield maximal Haagerup subalgebras. Our techniques are on one hand based on group-theoretic considerations, and on the other on certain results on intermediate von Neumann algebras, in particular these allowing us to deduce that all the intermediate algebras for certain inclusions arise from groups or from group actions. Some remarks and examples concerning maximal non-(T) subgroups and subalgebras are also presented, and we answer two questions of Ge regarding maximal von Neumann subalgebras.

33 pages; v5 introduces several minor changes and updates the references. Accepted for publication in the Groups, Geometry and Dynamics

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Cited by in corpus (4)

• Some Applications of Group Theoretic Rips Constructions to the Classification of von Neumann Algebras
• Maximal von Neumann subalgebras arising from maximal subgroups
• Maximal Haagerup subalgebras in $L({\mathbb{Z}}^2⋊S{L}_2(\mathbb{Z}))$
• Invariant $C^{\ast }$-subalgebras of the reduced group $C^{\ast }$-algebra