On a Fourier Expansion In Continuous Crossed Products

Abstract

Let $(M,R,\alpha )$ be a separable continuous $W^{\ast }$-dynamical system such that $M$ is $R$-finite.

Then any element in the crossed product $R{\otimes }_{\alpha }M$ of $M$ by $\alpha$ can be expressed as a vector valued tempered distribution $D^q{T}_{\eta }$ which is a weak${}^{\ast }$ limit of $T_{\xi }$, $\xi \in K(R;\mathfrak{B})$ in the dual space $S_U(R;\eta {)}^{\ast }$ of a generalized Schwartz space $S_U(R;\eta )$, where $K(R;\mathfrak{B})$ is the Tomita algebra corresponding to $R{\otimes }_{\alpha }M$.