Let $X$ be a locally compact Hausdorff topological space, denote by $M\_n$ the $C^\*$-algebra of complex $n\times n$ matrices, by $C\_0(X,M\_n)$ the $C^\*$-algebra of continuous functions on $X$ with values in $M\_n$ vanishing at infinity, and by $C\_b(X,M\_n)$ the $C^\*$-algebra of bounded continuous functions on $X$ with values in $M\_n$. Does there exist a description of all sub $C^\*$-algebras of $C\_0(X,M\_n)$ or $C\_b(X,M\_n)$? I am most interested in the (easier?) case where $X$ is compact and sub $C^\*$-algebras of $C(X,M\_n)$ containing the unit of $C(X,M\_n)$.


For the case $n=1$, i.e., for $C\_0(X)$, you can find the answer here: [What is the commutative analogue of a C\*-subalgebra?](https://mathoverflow.net/questions/35507/what-is-the-commutative-analogue-of-a-c-subalgebra).



What are the sub $C^*$-algebras of $C(X,M_n)$?
Every $C^\*$-subalgebra $A$ of $C\_0(X,M\_n)$ has irreps of dimension $\leq n.$ (Just because every irrep of $A$ can be continued to an irrep of $C\_0(X,M\_n)$). Such $C^\*$-algebras are called $n$-subhomogeneous. 


There is a complete (rather complicated) description of such $C^\*$-algebras in algebraic topological terms: 


Vasilʹev, N. B. "$C^∗$-algebras with finite-dimensional irreducible representations."
Uspehi Mat. Nauk 21 1966 no. 1 (127), 135–154. 


Much easier is to describe the $C^\*$-algebras with all irreps of the same dimension $n$. They are exactly the algebras of $C\_0$-sections of vector bundles over l.-c. hausdorff spaces with fiber $M\_n$ and group $PU\_n,$ see


Tomiyama, Jun; Takesaki, Masamichi
Applications of fibre bundles to the certain class of $C^∗$-algebras.
Tôhoku Math. J. (2) 13 1961 498–522. 


