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Weakly Radon–Nikodým Boolean algebras and independent sequences

Volume 241 / 2018

Antonio Avilés, Gonzalo Martínez-Cervantes, Grzegorz Plebanek Fundamenta Mathematicae 241 (2018), 45-66 MSC: Primary 03G05, 06E15, 28A60; Secondary 46B22, 46B50. DOI: 10.4064/fm404-5-2017 Published online: 13 October 2017

Abstract

A compact space is said to be weakly Radon–Nikodým (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 non-WRN 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. Rodríguez and R. Haydon concerning measures and the existence of nontrivial convergent sequences on WRN compacta, obtaining partial results on some natural subclasses.

Authors

  • Antonio AvilésDepartamento de Matemáticas
    Facultad de Matemáticas
    Universidad de Murcia
    30100 Espinardo, Murcia, Spain
    e-mail
  • Gonzalo Martínez-CervantesDepartamento de Matemáticas
    Facultad de Matemáticas
    Universidad de Murcia
    30100 Espinardo, Murcia, Spain
    e-mail
  • Grzegorz PlebanekInstytut Matematyczny
    Uniwersytet Wrocławski
    Pl. Grunwaldzki 2/4
    50-384 Wrocław, Poland
    e-mail

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