Completely Compact Herz-Schur Multipliers of Dynamical Systems

Journal article, 2022

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 (sic)(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 (sic)(r,alpha) G and provide a description of this class in the case of the irrational rotation algebra.