  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.
