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Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups. (English) Zbl 1342.46055
The authors develop a new approach, via the centralizers of certain states on the reduced \(C^*\)-algebras, to study maximal amenable von Neumann subalgebras coming from maximal amenable subgroups. Let \(\Lambda\subset \Gamma\) be a countable discrete group. \(\Lambda\) is said to be singular in \(\Gamma\) if there is a continuous action of \(\Gamma\) on some compact Hausdorff space \(X\) such that, for any \(\mu \in \mathrm{Prob}_{\Lambda}(X)\) and \(g\in \Gamma \backslash \Lambda\), we have \(g\cdot \mu \perp \mu\). The authors show that, if \(\Lambda\) is singular in \(\Gamma\), then \(L\Lambda\) is maximal amenable inside \(L\Gamma\). Many examples are given as applications of this result. In particular, for \(n\geq 2\), if \(\Gamma=SL_n(\mathbb{Z})\) and \(\Lambda\subset \Gamma\) is the subgroup of upper triangular matrices, then \(\Lambda\) is singular in \(\Gamma\).
The paper also gives many equivalent characterizations of singularity, and a simple proof for showing strong solidity for some group von Neumann algebras.

MSC:
46L10 General theory of von Neumann algebras
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
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