Constructing unconditional finite dimensional decompositions (Q1813350)

A fundamental problem in Banach space theory is to determine which properties of a Banach space are inherited by its complemented subspaces. Reflexivity, for example, is such a property. On the other hand, \textit{S. Szarek} [Acta Math. 159, 81-98 (1987; Zbl 0637.46013)], has shown that there exists a complemented subspace of a Banach space with a basis which fails to have a basis. In this paper, the authors show that complemented subspaces of a Banach space with unconditional finite dimensional decompositions satisfying certain extra assumptions also have unconditional finite dimensional decompositions.