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.

arXiv comment11 pages, minor revision, to appear in J. Math. Soc. Japan

References in corpus (10)

• Non commutative Lp spaces without the completely bounded approximation property
• Examples of groups which are not weakly amenable
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Cited by in corpus (2)

• The approximation property for locally compact quantum groups
• Crossed product splitting of intermediate operator algebras via 2-cocycles