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.

A non-commutative Fejér theorem for crossed products, the approximation property, and applications

“…Remark 5.2. The result of Theorem 5.1 also appears in the completely independent very recent work of Suzuki [37], of which we became aware after obtaining this result.…”

“…It was recently, independently, shown by Suzuki that for any C * -dynamical system (A, G, α) where G has the AP, G ⋊ A has the SOAP if and only if A has the SOAP [37,Theorem 3.6]. In particular, he obtains a different proof that C * λ (G) has the SOAP for any locally compact group G with the AP.…”

Exactness and SOAP of Crossed Products via Herz--Schur multipliers

“…Dynamical systems involving the action of a group with the AP have recently been studied by Crann-Neufang [3] and Suzuki [15]. Suzuki shows that if a locally compact group G has the AP then, for any C * -dynamical system (A, G, α), A ⋊ α,r G has the (S)OAP if and only if A has the (S)OAP.…”