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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?.

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I don't think there is a simple description: you can have $C^*$-subalgebras with non-Hausdorff spectra (simplest example: continuous functions $[0,1]\rightarrow M_n$ which are diagonal at $0$); and if you assume your $C^*$-subalgebra to be homogeneous (i.e. all its irreducible rep's are of the same dimension), then you have the $K$-theory of $C(X)$ showing up (if $p$ is a self-adjoint idempotent in $M_n(C(X))$, look at the subalgebra $pM_n(C(X))p$). – Alain Valette Mar 6 '13 at 20:10
up vote 6 down vote accepted

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.

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