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 C0(L) spaces and abstract M-spaces. Next, we prove that Wood's conjecture on almost transitivity of the norm in C0(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.