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.

Theorems & Definitions (27)

• Theorem
• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9