The point-wise closure of the space of continuous functions

Question

Let $X$ be a locally compact space and consider $C_0(X)$.

We denote $b(X)$, by the set of all bounded functions on $X$. It is easy to be checked that $b(X)$ may be considered as a C*-sub algebra of $B(\ell^2(X))$ via:

$$\gamma:b(X)\to B(\ell^2(X)): \gamma(\phi)f=\phi f$$

$\gamma(b(X))$ is also a von Neumann algebra. It means that $b(X)$ is itself a W*-algebra.

Q: What is the weak-star closure of $C_0(X)$ in $b(X)$?

Answer 1

The point-wise closure of $C_0(X)$ is $b(X)$. Indeed, choose a function $f\in b(X)$. For each finite subset $M\subset X$ let $f_M$ be a compactly supported continuous function which agrees with $f$ on $M$ (such a function exists by the Urysohn lemma for locally compact spaces). The net $(f_M)_{M}$ converges to $f$ point-wise.