The Szlenk index and local l_1-indices

TL;DR

This paper introduces new local l_1-indices for Banach spaces, establishes their relationship with the Szlenk index, and computes specific indices for certain spaces, advancing the understanding of Banach space geometry.

Contribution

It defines the l_1^+-index and l_1^+-weakly null index, relates them to the Szlenk index, and calculates these indices for specific Banach spaces.

Findings

l_1^+-weakly null index equals Szlenk index for spaces without l_1

l_1^+-weakly null index takes values omega^alpha for countable indices

I(C(omega^{omega^alpha}))=omega^{1+alpha+1}

Abstract

We introduce two new local l_1-indices of the same type as the Bourgain l_1 index; the l_1^+-index and the l_1^+-weakly null index. We show that the l_1^+-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain l_1. The l_1^+-weakly null index has the same form as the Bourgain l_1 index: if it is countable it must take values omega^alpha for some alpha<omega_1. The different l_1-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its local l_1-index, via the l_1^+-weakly null index. We show that I(C(omega^{omega^alpha}))=omega^{1+alpha+1}.