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? 



Crossed product of a non unital C*-algebra
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>


