TOWARDS A CHARACTERIZATION OF THE PROPERTY OF LEBESGUE

Abstract

There are three main contributions in this work. First, the proof that every stabilized asymptotic-${\ell }_1$ Banach space has the Property of Lebesgue is generalized to the coordinate-free case. Second, the proof that every Banach space with the Property of Lebesgue has a unique ${\ell }_1$ spreading model is generalized to cover a particular class of asymptotic models. Third, a characterization of the Property of Lebesgue is derived that applies to those Banach spaces with bases that admit in a strong sense favorable block bases. These results are significant because they demonstrate not only the efficacy of characterizing the Property of Lebesgue in terms of a connection between the local and the global asymptotic structures of certain Banach spaces, but also the possibility of finding a more general characterization in similar terms.