Completely compact Herz-Schur multipliers of dynamical systems

Abstract

We prove that if G is a discrete group and (A, G, α) is a C*-dynamical system such that the reduced crossed product A ⋊r,α G possesses property (SOAP) then every completely compact Herz-Schur (A, G, α)-multiplier can be approximated in the completely bounded norm by Herz-Schur (A, G, α)-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 ⋊r,α G and provide a description of this class in the case of the irrational rotation algebra.