Exel's approximation property and the CBAP of crossed products

Abstract

Let $\Gamma$ be a countable discrete group that acts on a unital separable $C^*$-algebra $A$ through an action $\alpha$. Suppose that the $C^*$-dynamical system $(A,\Gamma,\alpha)$ has Exel's approximation property, then $A$ has the CBAP if and only if the reduced crossed product $A\rtimes_{\alpha,r} \Gamma$ has the CBAP.