Invariant subalgebras rigidity for von Neumann algebras of groups arising as certain semidirect products

TL;DR

This paper investigates the rigidity of invariant subalgebras in von Neumann algebras associated with certain semidirect product groups, revealing new examples of amenable groups with unique invariant subalgebra properties.

Contribution

It provides the first example of an amenable group with the ISR property that has a non-trivial abelian normal subgroup and explores invariant subalgebras in lamplighter groups.

Findings

First example of an amenable group with ISR and a non-trivial abelian normal subgroup

An infinite amenable group with a unique invariant von Neumann subalgebra not from a normal subgroup

Analysis of invariant subalgebras in the lamplighter group's von Neumann algebra

Abstract

We study the ISR (von Neumann invariant subalgebra rigidity) property for certain discrete groups arising as semidirect products from algebraic actions on certain 2-torsion groups, mostly arising as direct products of $\mathbb{Z}_2$. We present, in particular, the first example of an amenable group with the ISR property that admits a non-trivial abelian normal subgroup. Several other examples are discussed, notably including an infinite amenable group whose von Neumann algebra admits precisely one invariant von Neumann subalgebra which does not come from a normal subgroup. We also investigate the form of invariant subalgebras of the group von Neumann algebra of the standard lamplighter group.