论文标题
$ L（\ Mathbb {z}^2 \ rtimes sl_2（\ Mathbb {z}））$ l（\ mathbb {z}^2 \ rtimes sl_2）中的最大haagerup subalgebras $
Maximal Haagerup subalgebras in $L(\mathbb{Z}^2\rtimes SL_2(\mathbb{Z}))$
论文作者
论文摘要
我们证明$ l（sl_2（\ textbf {k}））$是$ l（\ textbf {k}^2 \ rtimes sl_2（\ textbf {k}）$的最大haagerup von neumann subalgebra。然后，我们展示如何修改证明以处理$ \ textbf {k} = \ mathbb {z} $。证明的关键步骤是对$ l（sl_2（\ textbf {k}））之间的所有中间von neumann subergebras的完整说明algebraic操作$ sl_2（\ textbf {k}）\ curvearrowright \ wideHat {\ textbf {k}^2} $，通过改进关系$ ϕ \ sim ϕ'$，其中$ ϕ $，其中$ ϕ $，$ ϕ $，$ ϕ y）：= ϕ（-x，-y）$ for hast $（x，y）\ in \ textbf {k}^2 $。作为副产品，我们显示$ l（psl_2（\ mathbb {q}））$是$ l^{\ infty}（y）\ rtimes psl_2（\ rtimes psl_2（\ mathbb {q}}）中的最大von neumann subalgebra。特别是，$ psl_2（\ mathbb {q}）\ curvearrowrowright y $是主要动作，即它不承认非平凡的商动作。
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 $\textbf{k}=\mathbb{Q}$. Then we show how to modify the proof to handle $\textbf{k}=\mathbb{Z}$. 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 $ϕ\sim ϕ'$, where $ϕ$, $ϕ'\in \widehat{\textbf{k}^2}$ and $ϕ'(x, y):=ϕ(-x, -y)$ 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, i.e. it admits no non-trivial quotient actions.
