A character approach to the ISR property

TL;DR

This paper introduces a character-based method to analyze the ISR property in von Neumann algebras, establishing new criteria and examples for groups with this rigidity property.

Contribution

It develops a character approach to study the ISR property, linking it to the non-factorizable regular character property and providing new examples of groups with ISR.

Findings

Non-factorizable regular character property implies ISR for certain groups

Approximately finite groups possess the ISR property

Construction of non-amenable groups with ISR and non-trivial amenable radical

Abstract

We develop a character approach to study the invariant von Neumann subalgebras rigidity property (abbreviated as the ISR property) introduced in Amrutam-Jiang's work. First, we introduce the non-factorizable regular character property for groups and show that this implies the ISR property for any infinite ICC groups with trivial amenable radical.Various examples are shown to have this property. Second, we apply known classification results on indecomposable characters to show approximately finite groups have the ISR property. Based on this approach, we also construct non-amenable groups with the ISR property while having non-trivial amenable radical or without the non-factorizable regular character property.