Journal of Operator Theory

Volume 95, Issue 1, Winter 2026 pp. 103-117.

On invariant subalgebras when the ISR property fails

Authors: Yongle Jiang (1) Ruoyu Liu (2)
Author institution: (1) School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
(2) School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China; Current address: Dalian Jinpu New District Gaochengshan Middle School, Dalian, 116100, China

Summary: 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 icc 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 $SL_2(\mathbb{Z})$.

DOI: http://dx.doi.org/10.7900/jot.2024mar28.2481
Keywords: invariant von Neumann subalgebras, Haagerup radical, amenable radical

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