Exact Groups, Induced Ideals, and Fell Bundles

Abstract

Given a C*-algebra B which is graded over a discrete group G we consider ideals of B which are invariant under the projections onto each of the grading subspaces. If G is exact and the standard conditional expectation of B is faithful we show that all such ideals are induced, i.e. are generated by their intersection with the unit fiber algebra. This result is derived from a generalization to the context of Fell bundles of a Theorem by Kirchberg and Wasserman according to which a group is exact if and only if its reduced C*-algebra is exact.