A note on the crossed product $\mathcal{R}(\mathcal{A},\alpha )$ associated with PSL₂(ℝ)

Abstract

Given the hyperbolic measure dxdy/y ² on the upper half plane ℍ, the rational actions of PSL₂(ℝ) on ℍ induces a continuous unitary representation α of this group on the Hilbert space L ²(ℍ, dxdy/y ²). Supposing that $\mathcal{A}$ = {M _f : f ∈ L ^∞ (ℍ, dxdy/y ²)}, we show that the crossed product $\mathcal{R}(\mathcal{A},\alpha )$ is of type I. In fact, the crossed product $\mathcal{R}(\mathcal{A},\alpha )$ is *-isomorphic to the von Neumann algebra $\mathcal{B}(L^2 (P,\nu ))\bar \otimes \mathcal{L}_K$, where $\mathcal{L}_K$ is the abelian group von Neumann algebra generated by the left regular representation of K.