Category measures, the dual of $C(K)^\delta$ and hyper-Stonean spaces

TL;DR

This paper characterizes the dual of the Dedekind completion of continuous functions on a compact space and provides a topological characterization of hyper-Stonean spaces using category measures.

Contribution

It offers a new topological description of hyper-Stonean spaces via the dual of $C(K)^ abla$ and connects order continuity properties with Oxtoby's category measures.

Findings

Characterization of the dual of $C(K)^ abla$

Topological description of hyper-Stonean spaces

Relation between order continuity and category measures

Abstract

For a compact Hausdorff space $K$, we give descriptions of the dual of $C(K)^\delta$, the Dedekind completion of the Banach lattice $C(K)$ of continuous, real-valued functions on $K$. We characterize those functionals which are $\sigma$-order continuous and order continuous, respectively, in terms of Oxtoby's category measures. This leads to a purely topological characterization of hyper-Stonean spaces.