THE CROSSED PRODUCT VON NEUMANN ALGEBRAS ASSOCIATED WITH $SL_2(\mathbb{R})$

Abstract

Let $\mathcal{A}$ be the abelian von Neumann subalgebra $\{M_{f}: f \in L^{\infty}(\mathbb{H},\mu_{r})\}$ of $\mathcal{B}(L^{2}(\mathbb{H},\mu_{r}))$, where $\mathbb{H}$ is the upper half plane and the measure $d\mu_{r} = dx dy/y^{2-r}$. For any integers $r \gt 1$, the linear fractional action of $SL_{2}(\mathbb{R})$ on $\mathbb{H}$ induces a continuous action $\alpha$ of $SL_{2}(\mathbb{R})$ on $\mathcal{A}$. It is shown that the crossed product $\mathcal{R}(\mathcal{A}, \alpha)$ of $\mathcal{A}$ under the action $\alpha$ of $SL_{2}(\mathbb{R})$ is *-isomorphic to $\mathcal{B}(L^{2}(P, 2dx dy/y^{3-2r})) \overline{\otimes} \mathcal{L}_{K}$, where $SL_{2}(\mathbb{R}) = PK$ is the Iwasawa decomposition of $SL_{2}(\mathbb{R})$. Thus $\mathcal{R}(\mathcal{A}, \alpha)$ is of type $\mathrm{I}$.