Maximal Haagerup subalgebras in $L({\mathbb{Z}}^2⋊S{L}_2(\mathbb{Z}))$

Abstract

We prove that $L(S{L}_2(\text{k}))$ is a maximal Haagerup von Neumann subalgebra in $L({\text{k}}^2⋊S{L}_2(\text{k}))$ for $\text{k}=\mathbb{Q}$. Then we show how to modify the proof to handle $\text{k}=\mathbb{Z}$. The key step for the proof is a complete description of all intermediate von Neumann subalgebras between $L(S{L}_2(\text{k}))$ and $L^{\infty }(Y)⋊S{L}_2(\text{k})$, where $S{L}_2(\text{k})\curvearrowright Y$ denotes the quotient of the algebraic action $S{L}_2(\text{k})\curvearrowright \widehat{{\text{k}}^2}$ by modding out the relation $\varphi \sim {\varphi }^{\prime }$, where $\varphi$, ${\varphi }^{\prime }\in \widehat{{\text{k}}^2}$ and ${\varphi }^{\prime }(x,y):=\varphi (-x,-y)$ for all $(x,y)\in {\text{k}}^2$. As a by-product, we show $L(PS{L}_2(\mathbb{Q}))$ is a maximal von Neumann subalgebra in $L^{\infty }(Y)⋊PS{L}_2(\mathbb{Q})$; in particular, $PS{L}_2(\mathbb{Q})\curvearrowright Y$ is a prime action, i.e. it admits no non-trivial quotient actions.

v2: simply formulas and calculations and minor corrections according to referee report, accepted to JOT

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• On a Class of Type II${}_1$ Factors with Betti Numbers Invariants
• Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems
• Maximal subgroups and von Neumann subalgebras with the Haagerup property
• Maximal von Neumann subalgebras arising from maximal subgroups
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Cited by in corpus (2)

• Maximal subgroups and von Neumann subalgebras with the Haagerup property
• Invariant $C^{\ast }$-subalgebras of the reduced group $C^{\ast }$-algebra