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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?

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Hm. If $X = \mathbb{Z}$ and the action is by left translations then you get the compact operators on $\ell^2(\mathbb{Z})$ (you should prove this as an exercise). Anyway, this is not very unital. –  t.b. Aug 21 '11 at 12:04
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An earlier version of this question was crossposted to MO. –  t.b. Aug 21 '11 at 23:24

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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?

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