## 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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