The order bidual of C(X) for a realcompact space

TL;DR

This paper extends the classical result about the bidual of C(X) for compact spaces to realcompact spaces, showing it is isomorphic to C(tilde X) for some realcompact tilde X, with the isomorphism preserving the algebraic structure.

Contribution

It proves that the order bidual of C(X) for a realcompact space X is isomorphic to C(tilde X) for some realcompact tilde X, generalizing the compact case.

Findings

The order bidual of C(X) is isomorphic to C(tilde X) for a realcompact tilde X.

The space tilde X is unique up to homeomorphism.

The isomorphism preserves the $f$-algebra structure.

Abstract

It is well known that the bidual of $\mathrm C(X)$ for a compact space $X$, supplied with the Arens product, is isometrically isomorphic as a Banach algebra to $\mathrm C(\tilde X)$ for some compact space $\tilde X$. The space $\tilde X$ is unique up to homeomorphism. We establish a similar result for realcompact spaces: The order bidual of $\mathrm C(X)$ for a realcompact space $X$, when supplied with the Arens product, is isomorphic as an $f$-algebra to $\mathrm C(\tilde X)$ for some realcompact space $\tilde X$. The space $\tilde X$ is unique up to homeomorphism.