Show that $C_{0}(X,A)$ is a sub space of $B(H)$ for some Hilbert space $H$

Question

Let $A$ be $C^{\ast}$- 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 $t\in X$).

I wanna show that $C_{0}(X,A)$ is a sub space of $B(H)$ for some Hilbert space $H$. How can I construct such $H$?

Any hints/ideas?

P.S: I am not interested in proving by checking axioms of $C^{\ast}-$algebra.

Answer 1

If $\mu$ is a Borel measure with full support on $X$ and $A\subset B(K)$, you can take $H=L^2(X,\mu;K)$ with the action given by multiplication as usual. Of course, if you want to prove that $C_0(X;A)$ is a $C^\ast$-algebra, it may be easier to just check the abstract axioms.

Here is a more explicit description of the representation on $H$. For $f\in C_0(X;A)$ let $\pi(f)$ be operator acting on $H$ by $\pi(f)g(x)=f(x)g(x)$. Here $g(x)\in K$ and $f(x)\in A\subset B(K)$. It is easy to check that $\pi$ is an injective $\ast$-homomorphism.