A Characterization of a Coaction Reduced to That of a Closed Subgroup

Abstract

It is shown that, for any coaction $\alpha$ of a locally compact group $K$ on a properly infinite von Neumann algebra $A$ and a closed subgroup $H$ of $K$, $\alpha$ is cocycle conjugate to a coaction which comes from a coaction of $H$ if and only if the dual action $\widehat{\alpha}$ is induced by an action of $H$. We also include applications of the result concerning almost periodic coactions and the ranges of 1-cocycles on measured equivalence relations.