# The approximation property and exactness of locally compact groups

## TL;DR

This paper proves that locally compact groups with the approximation property (AP) have their reduced group C*-algebras possess the strong operator approximation property (SOAP), and that AP implies exactness for such groups.

## Contribution

It extends the theorem of Haagerup and Kraus to the C*-algebra context and establishes that AP implies exactness for general locally compact groups.

## Key findings

- AP implies SOAP for reduced C*-crossed products
- Reduced group C*-algebras of AP groups have SOAP
- AP implies exactness for 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.