The crossed product of a non unital C*-algebra

Question

Let $X$ be a locally compact space, and let $\mathbb{Z}$ act on $C_0(X)$ by an automorphism $\alpha$. Is the resulting crossed product unital?

Answer 1

Here is also a more general fact. Let $G$ act freely and properly on a space X. Then there is a (in general noncanonical) isomorphism $C_0(X) \rtimes_{r} G \cong C_0(X/G)\otimes \mathcal{K}(\ell^2(G))$ which implies nonunitality. Since the reduced crossed product should be a quotient of the full crossed product (is this true?) this also tells about unitality of the crossed product. Unfortunately I do not have a good reference for such statements. Does anyone?