Suppose a countable discrete amenable group $G$ acts continuously on a 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.

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.

Suppose an countable discrete amenable group $G$ acts continuously on a Compact Hausdorff space $X$, i.e. $\alpha:G\curvearrowright X$, which 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.

Suppose a countable discrete amenable group $G$ acts continuously on a 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.