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

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.