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Examples of mixing subalgebras of von Neumann algebras and their normalizers. (English) Zbl 1266.46046

Summary: We discuss different mixing properties for triples of finite von Neumann algebras \(B\subset N\subset M\), and we introduce families of triples of groups \(H<K<G\) whose associated von Neumann algebras \(L(H)\subset L(K)\subset L(G)\) satisfy \(\mathcal{N}_{L(G)}(L(H))''=L(K)\). It turns out that the latter equality is implied by two conditions: the equality \(\mathcal{N}_G(H)=K\) and the above mentioned mixing properties. Our families of examples also allow us to exhibit examples of pairs \(H<G\) such that \(L(\mathcal{N}_G(H))\not=\mathcal{N}_{L(G)}(L(H))''\).

MSC:

46L10 General theory of von Neumann algebras
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L55 Noncommutative dynamical systems
37A55 Dynamical systems and the theory of \(C^*\)-algebras
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