On maximal injective subalgebras in a \(w\Gamma\) factor
type: Item
instance of: scholarly article
title: On maximal injective subalgebras in a \(w\Gamma\) factor
zbMATH Open document ID: 1187.46045
published in: Science in China. Series A
publication date: 2009-12-07T00:00:00Z
review text: Let \(\mathcal{L}(F_{\mathbb{Q}})\rtimes_{\alpha}\mathbb{Z}\) be the crossed product von Neumann algebra of the free group factor \(\mathcal{L}(F_{\mathbb{Q}})\), associated with the left regular representation \(\lambda\) of the free group \(F_{\mathbb{Q}}\) with the set \(\{u_r:r\in\mathbb{Q}\}\) of generators, by an automorphism \(\alpha\) defined by \(\alpha(\lambda(u_r))=\exp(2\pi r i)\lambda(u_r)\), where \(\mathbb{Q}\) is the set of rational numbers. The author shows that \(\mathcal{L}(F_{\mathbb{Q}})\rtimes_{\alpha}\mathbb{Z}\) is a \(w\Gamma\) factor, and for each \(r \in \mathbb{Q},\) the von Neumann subalgebra \(\mathcal{A}_r\) generated in \(\mathcal{L}(F_{\mathbb{Q}})\rtimes_{\alpha}\mathbb{Z}\) by \(\lambda(u_r)\) and \(v\) is maximal injective, where \(v\) is the unitary implementing the automorphism \(\alpha\). In particular, \(\mathcal{L}(F_{\mathbb{Q}})\rtimes_{\alpha}\mathbb{Z}\) is a \(w\Gamma\) factor with a maximal abelian selfadjoint subalgebra \(\mathcal{A}_0\) with cannot be contained in any hyperfinite type II\(_1\) subfactor of \(\mathcal{L}(F_{\mathbb{Q}})\rtimes_{\alpha}\mathbb{Z}\), This gives a counterexample to Kadison's problem in the case of a \(w\Gamma\) factor.
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 5643352
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: maximal injective subalgebra
zbMATH Keywords: crossed product
zbMATH Keywords: \(w\Gamma \) factor
author: Chengjun Hou
reviewed by: Sh. A. Ayupov
MaRDI profile type: Publication
full work available at URL: s11425-008-0059-2
OpenAlex ID: W2068377340
Wikidata QID: Q114222431
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DOI: 10.1007/S11425-008-0059-2
description: scientific article; zbMATH DE number 5643352