Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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
An earlier version of this question was crossposted to MO. – t.b. Aug 21 '11 at 23:24

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?

share|cite|improve this answer
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. – cyc Mar 30 '15 at 3:50

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.