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.
Highlights
The notion of a Schur multiplier has its origins in the work of I
Grothenieck in [8], Schur multipliers can be identified with ∞ ⊗eh ∞, the extended Haagerup tensor product of two copies of ∞
In [10], McKee, Todorov and Turowska generalized the notion of Schur multipliers and Herz-Schur multipliers to the C∗-algebra valued case: a class of Schur
Summary
The notion of a Schur multiplier has its origins in the work of I. In [10], McKee, Todorov and Turowska generalized the notion of Schur multipliers and Herz-Schur multipliers to the C∗-algebra valued case: a class of Schur. A-multipliers and a class of Herz-Schur multiplier of semidirect product bundle are identified, where A is a C∗algebra faithfully represented on a Hilbert space H. In this ‘operator-valued’ case, the starting point is a function φ defined on the direct product X × Y , where X and Y are standard measure space, and taking values in the space CB(A, B(H)) of all completely bounded maps from A into the C∗-algebra O(H) of all bounded linear operators on H. Stinpring’s Theorem in a proposition which will be very important in Section 6 but with an easier proof; in Section 5 we study the generalization of Herz-Schur multipliers in the context of Fell bundles, including the generalized transference theorem between Schur multipliers and Herz-Schur multipliers (see e.g [10, Theorem 3.8]); in Section 6 we study the problem concerning the nuclearity of the reduced C∗-algebra by aid of the notion of the generalized Herz-Schur multipliers
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