The order Bidual of C(x) for a real compact space

Abstract

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