# Journal of Operator Theory

Volume 86, Issue 1, Summer 2021  pp. 203-230.

Maximal Haagerup subalgebras in $L(\mathbb{Z}^2\rtimes SL_2(\mathbb{Z}))$

Authors:  Yongle Jiang
Author institution: School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

Summary: We prove that $L(SL_2(\textbf{k}))$ is a maximal Haagerup--von Neumann subalgebra in $L(\textbf{k}^2\rtimes SL_2(\textbf{k}))$ for $\textbf{k}=\mathbb{Q}$ and $\textbf{k}=\mathbb{Z}$. The key step for the proof is a complete description of all intermediate von Neumann subalgebras between $L(SL_2(\textbf{k}))$ and $L^{\infty}(Y)\rtimes SL_2(\textbf{k})$, where $SL_2(\textbf{k})\curvearrowright Y$ denotes the quotient of the algebraic action $SL_2(\textbf{k})\curvearrowright \widehat{\textbf{k}}^2$ by modding out the relation $\phi\sim \phi'$, where $\phi$, $\phi'\in \widehat{\textbf{k}}^2$ and $\phi'(x, y):=\phi(-x, -y)$ for all $(x, y)\in \textbf{k}^2$. As a by-product, we show $L(PSL_2(\mathbb{Q}))$ is a maximal von Neumann subalgebra in $L^{\infty}(Y)\rtimes PSL_2(\mathbb{Q})$; in particular, $PSL_2(\mathbb{Q})\curvearrowright Y$ is a prime action.

DOI: http://dx.doi.org/10.7900/jot.2020mar09.2282
Keywords: Haagerup property, maximal Haagerup-von Neumann subalgebras, maximal von Neumann subalgebras, prime actions