Spaces not containing $\ell_1$ have weak aproximate fixed point property

TL;DR

The paper proves that certain convex subsets of Banach spaces lacking $ ext{ell}_1$ sequences have the weak approximate fixed point property, providing a new characterization of such Banach spaces through weak topology.

Contribution

It establishes that convex subsets without $ ext{ell}_1$ sequences possess the weak approximate fixed point property, offering a novel characterization of Banach spaces excluding $ ext{ell}_1$.

Findings

Convex subsets without $ ext{ell}_1$ sequences have the weak approximate fixed point property.

Characterization of Banach spaces not containing $ ext{ell}_1$ via weak topology.

Provides insights into fixed point properties in Banach spaces.

Abstract

A nonempty closed convex bounded subset $C$ of a Banach space is said to have the weak approximate fixed point property if for every continuous map $f:C\to C$ there is a sequence $\{x_n\}$ in $C$ such that $x_n-f(x_n)$ converge weakly to 0. We prove in particular that $C$ has this property whenever it contains no sequence equivalent to the standard basis of $\ell_1$. As a byproduct we obtain a characterization of Banach spaces not containing $\ell_1$ in terms of the weak topology.