Herz-Schur multipliers of dynamical systems

TL;DR

This paper generalizes Herz-Schur multipliers to non-commutative dynamical systems, linking them with Schur $A$-multipliers and providing a framework for their characterization and transfer in this broader setting.

Contribution

It introduces a new class of Herz-Schur multipliers for dynamical systems and establishes their relation to Schur $A$-multipliers, extending classical concepts.

Findings

Characterization of Schur $A$-multipliers

Identification of Herz-Schur multipliers with invariant Schur $A$-multipliers

Application to abelian groups and their duals

Abstract

We extend the notion of Herz-Schur multipliers to the setting of non-commutative dynamical systems: given a C*-algebra $A$, a locally compact group $G$, and an action $\alpha$ of $G$ on $A$, we define transformations on the (reduced) crossed product $A\rtimes_{r,\alpha} G$ of $A$ by $G$, which, in the case $A = \mathbb{C}$, reduce to the classical Herz-Schur multipliers. We also introduce a class of Schur $A$-multipliers, establish its characterisation which generalise the classical descriptions of Schur multipliers and present a transference theorem in the new setting, identifying isometrically the Herz-Schur multipliers of the dynamical system $(A,G,\alpha)$ with the invariant part of the Schur $A$-multipliers. We discuss special classes of Herz-Schur multipliers, in particular, those which are associated to a locally compact abelian group $G$ and its canonical action on the $C^*$-algebra $C^*(\Gamma)$ of the dual group $\Gamma$.