Approximation property of $C^{\ast }$-algebraic Bundles

Abstract

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

13 pages, Latex