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?
The crossed product of a non unital C*-algebra
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functional-analysis
noncommutative-geometry
c-star-algebras
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1Hm. 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. – 2011-08-21
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1An 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