On the comparison of norms of convolutors associated to 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 $(\pi \_A^{\alpha },\pi \_B^{\alpha })$ of unitary representations of $G$. The purpose of this paper is to provide results allowing to compare the norms of the operators $\pi \_A^{\alpha }(\mu )$ and $\pi \_B^{\alpha }(\mu )$ for bounded measures $\mu$ 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 $(\pi \_A^{\alpha },\pi \_B^{\alpha })$. Second, under amenability assumptions, to obtain transference of inequalities that will be useful in noncommutative ergodic theory.

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