Compact spaces associated to separable Banach lattices

TL;DR

This paper investigates the class of compact spaces arising as structure spaces of separable Banach lattices, revealing measure-theoretic and topological properties, especially in nonmetrizable cases, and highlighting open questions.

Contribution

It characterizes the compacta associated with separable Banach lattices, showing they admit specific measures and contain saturated copies of , and discusses open problems.

Findings

Every such compactum admits a strictly positive regular Borel measure of countable type that is analytic.

In nonmetrizable cases, these compacta are saturated with copies of .

Open questions remain about the full characterization of these compact spaces.

Abstract

We study the class of compact spaces that appear as structure spaces of separable Banach lattices. In other words, we analyze what $C(K)$ spaces appear as principal ideals of separable Banach lattices. Among other things, it is shown that every such compactum $K$ admits a strictly positive regular Borel measure of countable type that is analytic, and in the nonmetrizable case these compacta are saturated with copies of $\beta\mathbb N$. Some natural questions about this class are left open.