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

Given a $C^{\ast }$-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 $A{⋊}_{\alpha ,r}G$. The resulting characterisations of exactness and SOAP of $A{⋊}_{\alpha ,r}G$ generalise the corresponding statements for the reduced group $C^{\ast }$-algebra.