Let $\theta=m/n$ and let $A_{\theta}$ be the rational rotation C$^{*}$-algebra with rotation angle $\theta$. I.e., $A_{\theta}=C^{*}(u,v)$, where $u$ and $v$ are unitaries such that $vu=e^{2\pi i \theta}uv$. I know from here that $A_{\theta}$ is not a full matrix algebra. Given that $A_{\theta}$ has irreducible representations only of degree $n$, is it true that $A_{\theta}$ is a cutdown of a full matrix algebra? I.e.,
Is there a space $C(X,M_{n}(\mathbb{C}))$ and a projection $p\in C(X,M_{n}(\mathbb{C}))$, such that $A_{\theta}$ is isomorphic to $pC(X,M_{n}(\mathbb{C}))p$?
Yes, this is true.
As $\theta$ is rational, it follows that $A_{\theta}$ is Morita equivalent to $C(\mathbb{T}^2)$, see for example The classification of rational rotation algebras- Brabanter.
However, two unital $C^*$-algebras are Morita equivalent if and only if each is a full corner of $n\times n$ matrices over the other, for some $n$, see $C^*$-algebras associated with irrational rotations- Rieffel, Proposition 2.1.
Thus $A_{m/n}\cong pM_k(C(\mathbb{T}^2))p$, for some $k\in \mathbb{N}$ and full prjection $p\in M_k(C(\mathbb{T}^2))$.