A Bourgain-like property of Banach spaces with no copies of $c_0$

TL;DR

This paper characterizes when Banach spaces contain copies of c0 using index theory and provides new proofs of classical theorems related to Banach space structure and series behavior.

Contribution

It introduces a Bourgain-like property for Banach spaces without c0 copies and offers novel proofs of key theorems in Banach space theory.

Findings

Characterization of c0 embeddings via indexes

New proofs of James Distortion theorem

New proofs of Bessaga-Pe{2}czynski theorem

Abstract

We give a characterization of the existence of copies of $c_{0}$ in Banach spaces in terms of indexes. As an application, we deduce new proofs of James Distortion theorem and Bessaga-Pe{\l}czynski theorem about weakly unconditionally Cauchy series.