On invariant subalgebras when the ISR property fails

TL;DR

This paper classifies all G-invariant von Neumann subalgebras in the group von Neumann algebra for a specific semi-direct product group, especially when the usual rigidity property fails, revealing unique maximal subalgebras.

Contribution

It provides the first classification of G-invariant von Neumann subalgebras for certain i.c.c. groups lacking the ISR property, expanding understanding of subalgebra structures.

Findings

Classified all G-invariant von Neumann subalgebras for G=Z^2⋊SL_2(Z).

Identified the unique maximal Haagerup G-invariant subalgebra in L(G).

Extended the analysis to groups without the ISR property.

Abstract

We classify all $G$-invariant von Neumann subalgebras in $L(G)$ for $G=\mathbb{Z}^2\rtimes SL_2(\mathbb{Z})$. This is the first result on classifying $G$-invariant von Neumann subalgebras in $L(G)$ for i.c.c. groups $G$ without the invariant von Neumann subalgebras rigidity property (ISR property for short) as introduced in Amrutam-Jiang's work. As a corollary, we show that $L(\mathbb{Z}^2\rtimes \{\pm I_2\})$ is the unique maximal Haagerup $G$-invariant von Neumann subalgebra in $L(G)$, where $I_2$ denotes the identity matrix in $SL_2(\mathbb{Z})$.