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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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    An earlier version of this question was [crossposted to MO](http://mathoverflow.net/questions/73322/crossed-product-of-a-non-unital-c-algebra).2011-08-21

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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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    I would also like to have a reference for this fact because I'm now learning about crossed products. I wonder whether there is one.2015-03-30