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Let $X$ be a locally compact space, and let $T:X\rightarrow X$ be a homeomorphism. Then \begin{align*} &\alpha:C_0(X)\rightarrow C_0(X)\\\ &\alpha(f)=f\circ T \end{align*} is an automorphism. Now we can form the crossed product $C_0(X)\rtimes_\alpha\mathbb{Z}$.

When is this C*-algebra simple? and is it nuclear? And do we have a description of traces over it?

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    $\begingroup$ What we know about this C*-algebra? is a little bit of a vague question. For example: we know that it's abelian if and only if $T$ is the identity. Does that count as an answer? By the way: in the case when $T=Id_X$, then the traces on $C_0(X)\rtimes_\alpha \mathbb Z$ correspond bijectively to probability measures on $X\times S^1$. $\endgroup$
    – André Henriques
    Commented Aug 21, 2011 at 12:54
  • $\begingroup$ Thanks for the answer. I wanted to understand this C*-algebra in order to understand the crossed product in the non discrete case. For example is it nuclear or simple. An example which will give a crossed product which is simple will be great. Probably i am asking too much but i will be thankful for any bit of information. $\endgroup$
    – Steve
    Commented Aug 21, 2011 at 14:43
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    $\begingroup$ Then I would recommend that you rephrase your question (use the "edit" button) and include your actual questions: "when it this C*-algebra simple?", "when it this C*-algebra nuclear?". $\endgroup$
    – André Henriques
    Commented Aug 21, 2011 at 18:59
  • $\begingroup$ Seconding Andre's recommendations, if this does not seem too "mechant" $\endgroup$
    – Yemon Choi
    Commented Aug 22, 2011 at 0:28

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As $C_0(X)$ is nuclear and $\mathbb{Z}$ is amenable, the crossed product $C_0(X)\rtimes\mathbb{Z}$ is nuclear. Now, I'm fairly sure (I'm far from my files) that there is a result due to Zeller-Meier (1968), stating that $C_0(X)\rtimes_\alpha\mathbb{Z}$ is simple if and only if the homeomorphism $\alpha$ is minimal (i.e. all its orbits are dense). There must be a proof in K. Davidson's book "C*-algebras by examples". Finally traces on $C_0(X)\rtimes\mathbb{Z}$ are in 1-1 correspondence with $\alpha$-invariant Radon probability measures on $X$.

EDIT (with thanks to Andr\'e): Of course I was too quick in the last sentence; I should have said that every trace on $C_0(X)\rtimes\mathbb{Z}$ RESTRICTS to an $\alpha$-invariant probability measure on $X$. For a full description of traces on a crossed product by an abelian group, see Corollary 4 in http://arxiv.org/pdf/1010.0600v1

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  • $\begingroup$ Dear Alain, you say that traces are in 1-1 correspondence with $\alpha$-invariant probability measures on $X$... Doesn't that contradict my first comment above? $\endgroup$
    – André Henriques
    Commented Aug 22, 2011 at 0:48

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