Geometrical implications of certain infinite-dimensional decompositions

Abstract

We investigate the connections between the "global" structure of a Banach space (i.e. the existence of certain finite and infinite dimensional decompositions) and the geometrical properties of the closed convex bounded subsets of such a space (i.e. the existence of extremal and other topologically distinguished points). The global structures of various—supposedly pathological— examples of Banach spaces constructed by R. C. James turn out to be more "universal" than expected. For instance James-tree-type (resp. James-matrix-type) decompositions characterize Banach spaces with the Point of Continuity Property (resp. the Radon-Nikodým Property). Moreover, the Convex Point of Continuity Property is stable under the formation of James-infinitely branching tree-type "sums" of infinite dimensional factors. We also give several counterexamples to various questions relating some topological and geometrical concepts in Banach spaces.