On the comparison of norms of convolutors associated with noncommutative dynamics

Abstract

To any action of a locally compact group $G$ on a pair $(A,B)$ of von Neumann algebras is canonically associated a pair $(π_{A}^{α}, π_{B}^{α})$ of unitary representations of $G$. The purpose of this paper is to provide results allowing to compare the norms of the operators $π_{A}^{α}(μ)$ and $π_{B}^{α}(μ)$ for bounded measures $μ$ on $G$. We have a twofold aim. First, to point out that several known facts in ergodic and representation theory are indeed particular cases of general results about $(π_{A}^{α}, π_{B}^{α})$. Second, under amenability assumptions, to obtain transference of inequalities that will be useful in noncommutative ergodic theory.