Unconditional bases in countable-ℒ₁ spaces

Abstract

In this paper, we show that a countably- L 1 {\mathcal {L}_1} space which has an unconditional basis is isomorphic to some echelon sequence space of order 1. As a consequence, a countably- L 1 {\mathcal {L}_1} space with a basis is nuclear if all its bases are unconditional (this gives a partial answer to a conjecture of Wojtynski). We also study those countably- L 1 {\mathcal {L}_1} spaces on which a Fréchet lattice structure can be defined.