${\ell }_1$ spreading models and FPP in Banach spaces with monotone Schauder basis

Abstract

The main result of this paper is a fixed point result relating the spreading model structure of Banach spaces and Schauder basis with not too large basis constant. As a striking consequence, we deduce that every super-reflexive space has the fixed point property thus solving a long-standing open question in metric fixed point theory.

The first statement on page 9 is not necessarily true. Roughly speaking, the problem is that the indices "i_s" and "r" are competing with each other and therefore what I believed to be immediate, as happens naturally in the case of a single index, and as can be seen in the proof of Theorem 6.7 of the FHHMZ reference, is in fact not immediate in the situation where double indices are involved