Completely Compact Herz–Schur Multipliers of Dynamical Systems

Abstract

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