Classification of Invariant Subalgebras in a class of factors with property (T)

TL;DR

This paper classifies all invariant von Neumann subalgebras in group factors associated with certain property (T) groups, revealing new structural insights and extending previous results for specific cases.

Contribution

It provides the first classification of invariant subalgebras for a broad class of property (T) groups with new structural results.

Findings

Classified all G_n-invariant subalgebras in L(G_n) for n≥2.

Established uniqueness of maximal Haagerup G_n-invariant subalgebra.

Extended classification results to groups with property (T) for n≥3.

Abstract

Let $n\geq 2$ and $G_n=\mathbb{Z}^n\rtimes SL_n(\mathbb{Z})$. We classify all $G_n$-invariant von Neumann subalgebras in $L(G_n)$. For $n=2$, this gives an alternative proof of the previous result of Jiang-Liu. For $n\geq 3$, this gives the first class of property (T) groups without the invariant subalgebras rigidity property but invariant subalgebras in the corresponding group factors can still be classified. As a corollary, $L(G_n)$ admits a unique maximal Haagerup $G_n$-invariant von Neumann subalgebra.