Transitivity of M-spaces and Wood's conjecture

Abstract

For a wide class of categories of Banach spaces, we show that the existence of an (almost) transitive element implies the existence of a separable almost transitive element. We give some applications to C 0 ( L ) spaces and abstract M -spaces. Next, we prove that Wood's conjecture on almost transitivity of the norm in C 0 ( L ) can be reduced to the case in which the one-point compactification of L is metrizable. We construct a simple example of transitive M -space and we show the existence of almost transitive separable M -spaces that are isomorphic to C [0, 1]. These M -spaces have a rich M -structure and they are counterexamples for several questions about centralizers.