When will the $G$-invariant measure space be isomophic to the tracial state space of the crossed product $C^\ast$-algebra

Question

Suppose a countable discrete amenable group $G$ acts continuously on a infinite Compact Hausdorff space $X$, i.e. $\alpha:G\curvearrowright X$. Suppose $\alpha$ is minimal. Write $M_G(X)$ for all $G$-invariant Borel Probability measures on $X$ and write $T(C(X)\rtimes_rG)$ be the tracial state space of the crossed product $C^\ast$-algebra $C(X)\rtimes_rG$. My question is that can we always identify $T(C(X)\rtimes_rG)$ with $M_G(X)$ by the formular $\tau_\mu(a)=\int_X E(a)d\mu$ where $E$ is the conditional expection from $C(X)\rtimes_rG$ to $C(X)$? That is, whether $\mu\rightarrow \tau_\mu$ is a bijiection between this two sets ( which will be a homeomorphism w.r.t weak*-topology)?

I know if $G=\mathbb{Z}$, then it holds. I guess it is a standard fact but I cannot find a proper reference for the general case.

Thank you for all helps.

Answer 1

Suppose $(G,\alpha, C(X))$ is a triple for which this is true. Then let $\beta$ be the action of $G\times \mathbb{Z}$ on $C(X)$ which agrees with $\alpha$ on $G$ and is trivial on $\mathbb{Z}$. The statement will no longer be true for this action because $X$ hasn't changed but the crossed product has been tensored with $C(\mathbb{T})$ so it has many more tracial states.

(Answer to the original question which omitted the hypothesis that $X$ is infinite: I don't think this is true for $G = \mathbb{Z}$. Let $X$ be a singleton and let $\alpha$ be the trivial action of $\mathbb{Z}$. Then $C(X)\rtimes \mathbb{Z} \cong C^*(\mathbb{Z}) \cong C(\mathbb{T})$, which is abelian so it has lots of tracial states, but there is only one probability measure on $X$.)