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.

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    $\begingroup$ Not an answer to your question but just a remark: if you endow ∗ with the 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
  • $\begingroup$ @user124775 Which topology would you like to use on $C(X)^*$ in order to define $C(X)^{**}$? If it is the weak-* topology, then you will get back $C(X)$ no matter what. So do you want the strong dual topology? $\endgroup$ – Robert Furber May 28 '18 at 0:48
  • $\begingroup$ @RobertFurber, it is customary to use the strong topology when discussing the second dual of a lcHs. Otherwise, everything is trivial as you say. $\endgroup$ – user124775 May 28 '18 at 8:32

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