Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full <I>C</I><SUP>*</SUP>-Algebras

Abstract

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.