$\ell_1$ spreading models and FPP in Banach spaces with monotone Schauder basis

TL;DR

This paper establishes a fixed point property for super-reflexive Banach spaces by linking spreading models and Schauder bases, solving a long-standing open problem in metric fixed point theory.

Contribution

It introduces a fixed point result connecting spreading models and Schauder bases with small basis constants in Banach spaces, proving all super-reflexive spaces have the FPP.

Findings

Super-reflexive Banach spaces have the fixed point property.

The fixed point result relates spreading models and Schauder bases.

Solves a long-standing open problem in metric fixed point theory.

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.