Multipliers and Duality for Group Actions

TL;DR

This paper introduces operator-valued multipliers related to group actions, explores their properties via duality, and connects them to classical Herz--Schur multipliers for abelian groups and their duals.

Contribution

It defines operator-valued Schur and Herz--Schur multipliers through module actions and establishes their properties using duality theory, linking to classical multipliers.

Findings

Operator-valued multipliers are characterized via module actions.

Properties of these multipliers follow from duality theory.

A subset of Herz--Schur multipliers corresponds to classical multipliers on product groups.

Abstract

We define operator-valued Schur and Herz--Schur multipliers in terms of module actions, and show that the standard properties of these multipliers follow from well-known facts about these module actions and duality theory for group actions. These results are applied to study the Herz--Schur multipliers of an abelian group acting on its Pontryagin dual: it is shown that a natural subset of these Herz--Schur multipliers can be identified with the classical Herz--Schur multipliers of the direct product of the group with its dual group.