The fixed point property via dual space properties

TL;DR

This paper investigates conditions under which Banach spaces have the fixed point property, linking dual space properties like weak* compactness and the Kadec-Klee property to fixed point existence.

Contribution

It establishes new criteria connecting dual space properties with the fixed point property in Banach spaces, including E-convex spaces.

Findings

Banach spaces with dual weak* sequential compactness have the weak fixed point property.

Dual space satisfying weak* uniform Kadec-Klee property implies fixed point property.

E-convex Banach spaces, including uniformly nonsquare spaces, possess the fixed point property.

Abstract

A Banach space has the weak fixed point property if its dual space has a weak$^*$ sequentially compact unit ball and the dual space satisfies the weak$^*$ uniform Kadec-Klee property; and it has the \fpp if there exists $\epsilon>0$ such that, for every infinite subset $A$ of the unit sphere of the dual space, $A\cup (-A)$ fails to be $(2-\epsilon)$-separated. In particular, $E$-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.