Let $A$ be $C^*$- Algebra and $X$ be a locally compact Hausdorff space and $C_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{\ast}(t)={f(t)}^{\ast}$ (for $t\in X$). It is well known that $C_0(X,A)$ is $C^{\ast}-$ Algebra. Let $x\in X$ and $\pi$ be a representation of $A$ then the map $\pi_x$ defined as $\pi_x(f)=\pi(f(x))$ is a representation of $C_0(X,A)$
Is there any characterisation of representations of $C_0(X,A)$ in terms of representations of $C_0(X)$ and representations of $A$?
I am guessing that identification of $C_0(X,A)$ in terms of tensor should help here. Any ideas?
Yes, using the tensor product of $C^* $-algebras tells us that any representation of $C_0(X,A)$ comes from representations of $C_0(X)$ and $A$. Since $C_0(X)$ is nuclear, there's only one tensor product norm on the algebraic tensor product $C_0(X) \otimes_{alg} A$. The completion under this norm is in fact $C_0(X,A)\cong C_0(X) \otimes A$. Suppose $\pi:C_0(X) \otimes A\to \mathcal{L}( \mathcal{H})$ is a (non-degenerate) representation of $C_0(X)\otimes A$. Then, we get unique (non-degenerate) representations $\pi_1: C_0(X) \to \mathcal{L}(\mathcal{H})$ and $\pi_2: A \to \mathcal{L}(\mathcal{H})$ such that for any $f \in C_0(X)$ and $a\in A$ $$ \pi(f \otimes a) = \pi_1(f)\pi_2(a)=\pi_2(a)\pi_1(f) $$ In fact, if $(f_\nu)_{\nu \in N}$ is an approximate identity for $C_0(X)$ and $(a_\lambda)_{\lambda \in \Lambda}$ an approximate identity for $A$, the operators $\pi_1(f)$ and $\pi_2(a)$ are strong limits of $\pi(f \otimes a_\lambda)$ and $\pi(f_\mu \otimes a)$ respectively.