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.

In this paper we develop the notion of Schur multipliers and Herz-Schur multipliers to the context of Fell bundle, as a generalization of the theory of multipliers of locally compact groups and crossed products. We prove a characterization theorem of this generalized Schur multiplier in terms of the representation of Fell bundles. In order to prove this characterization theorem we define a new class of completely bounded maps; and discuss in detail of its properties. In this process, by the way, we give a new proof of Stinpring's Theorem of non-unital version. Then we investigate the transference theorem of Schur multipliers and Herz-Schur multipliers, which is a generalization of the transference theorem well-known either in the group case or crossed products. We use the notion of multipliers to define an approximation property of Fell bundles. Then we give a necessary and sufficient condition if the reduced cross-sectional algebra of a Fell bundle over a discrete groups is nuclear in terms of this generalized notion. This is a generalization of the classical theorem concerning the amenability of locally compact groups. As an application, we prove that for a Fell bundle, if its cross-sectional algebra is nuclear, then for any subgroup of the group on which the Fell bundle is defined, the cross-sectional algebra of the restricted Fell bundle on this subgroup is nuclear.

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.