Let $X$ be a finite set of cardinality $n$. Consider a minimal action of $\mathbb{Z}$ on $X$ which is nothing but a permutation $\sigma$ of $X$. Can I get a reference or proof for the fact:
The cross product $C(X) \rtimes_\sigma \mathbb{Z}$ is isomorphic to $M_n(C(\mathbb{T }))$.
Fix $x\in X$. Since $\sigma$ is minimal, the stabilizer $\{k\in\mathbb{Z}\mid \sigma^k(x)=x\}$ is $n\mathbb{Z}$ and the action induced by $\sigma$ is equivalent to the action by left translation on $\mathbb{Z}/n\mathbb{Z}$. Now you can appeal to Green's imprimitivity theorem, which implies $C(\mathbb{Z}/n\mathbb Z)\rtimes_\sigma \mathbb{Z}\cong C^\ast(n\mathbb{Z})\otimes M_n(\mathbb{C})\cong M_n(C(\mathbb{T}))$.
Luckily, $\mathbb{Z}$ is discrete and countable, which makes the proof considerably simpler, but it's still a bit lengthy, so I just refer to Example 10.11 in these lecture notes.