Exactness and SOAP of Crossed Products via Herz--Schur multipliers

Abstract

Given a $C^*$-dynamical system $(A,G,\alpha)$, with $G$ a discrete group, Schur $A$-multipliers and Herz--Schur $(A,G,\alpha)$-multipliers are used to implement approximation properties, namely exactness and the strong operator approximation property (SOAP), of $A \rtimes_{\alpha , r} G$. The resulting characterisations of exactness and SOAP of $A \rtimes_{\alpha , r} G$ generalise the corresponding statements for the reduced group $C^*$-algebra.