Weakly Radon-Nikod\'ym Boolean algebras and independent sequences

TL;DR

This paper explores the properties of weakly Radon-Nikodým Boolean algebras, their relation to minimally generated algebras, and constructs examples that answer open questions about their structure and measure-theoretic properties.

Contribution

It demonstrates the incomparability of WRN and minimally generated algebras and constructs a specific non-WRN algebra with a separable Rosenthal compactum, addressing open questions.

Findings

WRN and minimally generated algebras are incomparable

Constructed a minimally generated non-WRN algebra with a separable Rosenthal compactum

Partial results on measures and convergent sequences in WRN compacta

Abstract

A compact space is said to be weakly Radon-Nikod\'{y}m (WRN) if it can be weak*-embedded into the dual of a Banach space not containing $\ell_1$. We investigate WRN Boolean algebras, i.e. algebras whose Stone space is WRN compact. We show that the class of WRN algebras and the class of minimally generated algebras are incomparable. In particular, we construct a minimally generated nonWRN Boolean algebra whose Stone space is a separable Rosenthal compactum, answering in this way a question of W. Marciszewski. We also study questions of J. Rodr\'{i}guez and R. Haydon concerning measures and the existence of nontrivial convergent sequences on WRN compacta, obtaining partial results on some natural subclasses.