Positive definiteness and Fell bundles over discrete groups

Abstract

We introduce a natural concept of positive definiteness for bundle maps between Fell bundles over (possibly different) discrete groups and describe several examples. Such maps induce completely positive maps between the associated full cross-sectional $C^{\ast }$-algebras in a functorial way. Under the assumption that the kernel of the homomorphism connecting the groups under consideration is amenable, they also induce completely positive maps between the associated reduced cross-sectional $C^{\ast }$-algebras. As an application, we define an approximation property for a Fell bundle over a discrete group which generalizes Exel's approximation property and still implies the weak containment property. Both approximation properties coincide when the unit fibre is nuclear.

28 pages. Somewhat shortened and slightly reorganized, according to the suggestions of a referee

References in corpus (10)

• Representation theory for subfactors, $\lambda$-lattices and C*-tensor categories
• Positive Herz-Schur multipliers and approximation properties of crossed products
• Noncommutative Cartan C*-subalgebras
• Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles
• Equivalence of Fell bundles over groups
• Aperiodicity: the almost extension property and uniqueness of pseudo-expectations
• Reduced C*-algebras of Fell bundles over inverse semigroups
• Amenability and approximation properties for partial actions and Fell bundles
• Pure infiniteness and ideal structure of $C^{\ast }$-algebras associated to Fell bundles
• Morita enveloping Fell bundles

Cited by in corpus (1)

• On the topological ranks of Banach ${}^{\ast }$-algebras associated with groups of subexponential growth