Subalgebras of certain C*-algebras

Question

Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and $m<=n$ ?

Answer 1

This is not even even true for a singleton $X=\{\ast\}$. In this case $C(X,M_n(\mathbb{C}))\simeq M_n(\mathbb{C})$. Consider, for instance, $A$ the subalgebra of diagonal matrices. Then $A\simeq \mathbb{C}^n$. So for $n\geq 2$, this is not isomorphic to any $C(Y,M_m(\mathbb{C}))$ for $Y\subseteq X$, as the latter are isomorphic to $M_m(\mathbb{C})\not\simeq \mathbb{C}^n$.