On maximal injective subalgebras in a \(w\Gamma\) factor
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.
scientific article; zbMATH DE number 5643352