Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for some compact Hausdorff space $Z$ (for example, it is commutative and satisfies the C*-condition, so the Gelfand-Naimark theorem applies). Alternatively, it may be represented as $L_\infty(\mu)$ for some non-separable measure $\mu$.

I was wondering if a similar descriptions are available when $X$ is only a **completely regular space** and $C(X)$ is considered under the **topology of uniform convergence on compact sets** (the compact-open topology). In this case, $C(X)$ is still a locally convex topological algebra. Is there any sensible description of $C(X)^{**}$ in this case?

Any references to the literature will be appreciated.

topology of uniform convergence on compact sets(similarly to $C(X)$), then $C(X)^{∗∗}=C(X)$ for each paracompact locally compact space $X$: en.wikipedia.org/wiki/Stereotype_algebra $\endgroup$ – Sergei Akbarov May 26 '18 at 6:21