Completely compact Herz-Schur multipliers of dynamical systems

TL;DR

This paper characterizes completely compact Herz-Schur multipliers in dynamical systems, showing they can be approximated by finite-rank multipliers under certain properties, with specific results for the irrational rotation algebra.

Contribution

It establishes approximation results for completely compact Herz-Schur multipliers in C*-dynamical systems with property (SOAP) and explores their structure in the irrational rotation algebra.

Findings

Completely compact Herz-Schur multipliers can be approximated by finite-rank multipliers.

If G has the approximation property, these multipliers coincide with the closure of A(G).

The class of invariant completely compact Herz-Schur multipliers is described for the irrational rotation algebra.

Abstract

We prove that if $G$ is a discrete group and $(A,G,\alpha)$ is a C*-dynamical system such that the reduced crossed product $A\rtimes_{r,\alpha} G$ possesses property (SOAP) then every completely compact Herz-Schur $(A,G,\alpha)$-multiplier can be approximated in the completely bounded norm by Herz-Schur $(A,G,\alpha)$-multipliers of finite rank. As a consequence, if $G$ has the approximation property (AP) then the completely compact Herz-Schur multipliers of $A(G)$ coincide with the closure of $A(G)$ in the completely bounded multiplier norm. We study the class of invariant completely compact Herz-Schur multipliers of $A\rtimes_{r,\alpha} G$ and provide a description of this class in the case of the irrational rotation algebra.