I am reading about Hilbert $C^*$-modules in the book written by E. Christopher Lance.
I've found there the following claim:
If $X$ is a locally compact Hausdorff space, then $L(C_0(X,H))=C_b^{str}(X,L(H)).$
(Where $str$ is the $SOT^*$-topology on $B(H)$, i.e. $T_n\rightarrow_{SOT^*} T$ if $T_n x\rightarrow Tx$ and $T_n^*x\rightarrow T^*x$).
I reduced the problem to showing that the multiplier algebra of $C_0(X,K(H))$ is $C_b^{SOT^*}(X,B(H))$
My attempt:
Represent $\alpha: C_0(X,K(H))\to B(L^2(X,H))$ by multiplication, that is $(\alpha_f\xi)(x)=f(x)\xi(x)$ for all $f\in C_0(X,K(H)), \xi \in L^2(X,H),x\in X$.
Now, it suffices to show that the idealiser of $\alpha( C_0(X,K(H)))$ is $C_b^{SOT^*}(X,B(H))$. If we denote the idealiser by $B$, it is defined as follows:
$B=\{T\in B(L^2(X,H)| \ T\alpha( C_0(X,K(H)))\subseteq \alpha( C_0(X,K(H))); \alpha( C_0(X,K(H)))T\subseteq \alpha( C_0(X,K(H)))\}$.
The easy direction is showing that $C_b^{SOT^*}(X,B(H))\subseteq B$, and still I'm not sure how to complete my proof:
I need to show that if $f\in C_0(X,K(H))$ and $g\in C_b^{SOT^*}(X,B(H))$ then $fg\in C_0(X,K(H))$ and so is $gf.$ They "vanish at infinity" by boundness of $g$ and the fact that $f$ vanishes. $g(x)f(x),f(x)g(x)\in K(H)$ since $K(H)$ is an ideal in $B(H)$. So, I just have to show that if $(x_{\lambda})_{\lambda\in \Lambda}$ is a net in $X$ that converges to some $x\in X$ then $f(x_{\lambda})g(x_{\lambda})$ converges to $f(x)g(x)$ in the operator norm.
Any help would be appreciated, Thank you.