Approximation property of $C^*$-algebraic Bundles

TL;DR

This paper investigates the approximation property of $C^*$-algebraic bundles over locally compact groups, showing conditions under which full and reduced cross-sectional $C^*$-algebras coincide and when nuclearity is preserved.

Contribution

It introduces the approximation property for $C^*$-algebraic bundles over locally compact groups and establishes its implications for the equality of full and reduced cross-sectional $C^*$-algebras and nuclearity.

Findings

Full and reduced cross-sectional $C^*$-algebras coincide under the approximation property.

Nuclearity is preserved in semi-direct product bundles with the approximation property.

Comparison between the approximation property and amenability for discrete groups.

Abstract

In this paper, we will define the reduced cross-sectional $C^*$-algebras of $C^*$-algebraic bundles over locally compact groups and show that if a $C^*$-algebraic bundle has the approximation property (defined similarly as in the discrete case), then the full cross-sectional $C^*$-algebra and the reduced one coincide. Moreover, if a semi-direct product bundle has the approximation property and the underlying $C^*$-algebra is nuclear, then the cross-sectional $C^*$-algebra is also nuclear. We will also compare the approximation property with the amenability of Anantharaman-Delaroche in the case of discrete groups.