TITLE: When is it $C(X)$?
QUESTION [5 upvotes]: Suppose that $\tilde{X}$ is a compact space. If $C(\tilde{X})$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that $C(\tilde{X})=C_0(X)^{**}$?
P.S. Note that the converse is always true, namely if $X$ is a locally compact space, there is a compact space $\tilde{X}$ such that $C_0(X)^{**}=C(\tilde{X})$. The Banach space $C_0(X)^{**}$ turns into a Banach algebra with the first (second) Arens product. This is a non-trivial product on $C_0(X)^{**}$.

REPLY [5 votes]: This is a detailed version of a previous comment that gives a partial answer. 
If $K$ is a compact space and $C(K)$ is isometrically isomorphic to $X^{**}$, then $X$ is a $\mathcal{L}_{\infty}$-space. Hence $X^{**}$ is injective. 
It was proved by R. Haydon [Israel J. Math. 31 (1978), 142-152] that an injective bidual space is isomorphic to $\ell_\infty(\Gamma)$ for some set $\Gamma$. Hence $C(K)$ is isomorphic to $c_0(\Gamma)^{**}$. 
I do not know if $C(K)$ is isometrically isomorphic to $c_0(\Gamma)^{**}$.