Examples of mixing subalgebras of von Neumann algebras and their normalizers

TL;DR

This paper explores mixing properties of triples of finite von Neumann algebras and their group-theoretic counterparts, revealing conditions under which normalizers coincide or differ, thus advancing understanding of algebraic and group structures.

Contribution

It introduces new mixing conditions for triples of von Neumann algebras and constructs examples illustrating when algebraic normalizers match or differ from group-theoretic normalizers.

Findings

Normalizers of subalgebras can be characterized by mixing properties.

Examples show normalizers in group algebras can differ from algebraic normalizers.

Certain conditions ensure the equality of algebraic and group-theoretic normalizers.

Abstract

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 $L(\mathcal{N}_G(H))\not=\mathcal{N}_{L(G)}(L(H))"$.