Dual space of the space of matrix-valued, continuous functions on a locally-compact, Hausdorff space vanishing at infinity. Let $X$ be a locally-compact, Hausdorff space, and let $\mathcal{A}:=C_{0}(X)$ the $C^{*}$-algebra of complex-valued, continuous functions vanishing at infinity.
Let $\mathcal{B}:=M_{n}(\mathbb{C})$ the $C^{*}$-algebra of $n\times n$ complex matrices.
The tensor product $\mathcal{A}\otimes\mathcal{B}$ is well-defined as a $C^{*}$-algebra and it is isomorphic to $C_{0}(X,\mathcal{B})$ (see for instance thm: II.9.4.4 in Blackadar's book).
Since the dual space $\mathcal{A}'$ of $\mathcal{A}$ may be identified with the space of complex-valued Radon measures on $X$, and the dual space $\mathcal{B}'$ may be identified with $\mathcal{B}$ itself, I was wondering if it is possible to identify the dual space $(\mathcal{A}\otimes\mathcal{B})'\cong(C_{0}(X,\mathcal{B}))'$ with the space of $\mathcal{B}$-valued Radon measures on $X$, and thus the space of states of $\mathcal{A}\otimes \mathcal{B}$ with the space of $S$-valued Radon measures on $X$, where $S$ is the space of positive semidefinite matrices with unit trace.
Since it is just a mere curiosity triggered by a casual conversation I had yesterday, I did not yet really try to prove anything, but I have the impression that this could be a well-known result for experts in $C^{*}$-algebra theory.
 A: It is true that if $A$ is any $C^*$-algebra, then $M_n(A)^*$ is isomorphic to $M_n(A^*)$. In your case, if $R_X$ denotes the space of Radon measures on $X$
\begin{align}
C_0\big(X,M_n(\mathbb C)\big)^*
& \cong (A\otimes B)^*
= \big(C_0(X)\otimes M_n(\mathbb C)\big)^* \\
& = M_n(C_0(X))^*
\cong M_n(C_0(X)^*)
\cong M_n(R_X).
\end{align}
Now I am no expert in measure theory but I don't feel confident about matrices with measure-entries being the same as measures with matrix values. That's only my guess though.
As requested: let $\zeta_{i,j}:A\to M_n(A)$ denote the map $\zeta_{i,j}(a)=a\otimes e_{i,j}$, where $e_{k,l}$ are matrix units (i.e. $a\otimes e_{i,j}$ is the matrix with $0$'s everywhere and $a$ in the i,j slot). This is easily seen to be a linear isometry. Define $T:M_n(A)^*\to M_n(A^*)$ by $T(\varphi)=[\zeta_{i,j}^*(\varphi)]_{i,j}\equiv[\varphi\circ\zeta_{i,j}]_{i,j}$. This is easily seen to be a vector space isomorphism. This is for example demonstrated in C. Lance's paper "On nuclear $C^*$-algebras" (1973), but the details are easy to fill in.
