Weakly Radon–Nikodým Boolean algebras and independent sequences
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.