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Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups. (English) Zbl 1369.46052

Given a II\(_1\) factor, it is natural to study the structure of its hyperfinite subalgebras. In the 1960s, Kadison addressed a general question: is any self-adjoint element in a II\(_1\) factor \(M\) contained in a hyperfinite subfactor of \(M\)? A first answer to this question was provided by S. Popa [Adv. Math. 50, 27–48 (1983; Zbl 0545.46041)], who showed that the von Neumann subalgebra of \(LF_n\) \((n\geq 2)\) generated by one of the generators of \(F_n\) is maximal amenable, and yet it is abelian.
In this article, the authors prove a general rigidity result on maximal amenability from a “group-subgroup” situation, to the “von Neumann algebra-subalgebra” they generate. The main focus is on hyperbolic groups. At the group level, infinite amenable subgroups of hyperbolic groups are completely understood: they are virtually cyclic, and they act in a nice way on the Gromov boundary of the group. Using this fact, they show the following, generalizing the mentioned result of Popa [loc.cit.].
Theorem A. Consider a hyperbolic group \(G\) and an infinite, maximal amenable subgroup \(H<G.\) Then the group von Neumann algebra \(LH\) is maximal amenable inside \(LG\).

MSC:

46L10 General theory of von Neumann algebras
46L55 Noncommutative dynamical systems
46L37 Subfactors and their classification

Citations:

Zbl 0545.46041
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References:

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