The approximation property and exactness of locally compact groups

Abstract

We extend a theorem of Haagerup and Kraus in the C^*-algebra context: for a locally compact group with the approximation property (AP), the reduced C^*-crossed product construction preserves the strong operator approximation property (SOAP). In particular their reduced group C^*-algebras have the SOAP. Our method also solves another open problem: the AP implies exactness for general locally compact groups.