Retractions and the bounded approximation property in Banach spaces

TL;DR

This paper investigates the conditions under which Banach spaces admit certain Lipschitz retracts, establishing that the bounded approximation property is necessary, thus addressing a question posed by Godefroy and Ozawa.

Contribution

It provides a partial answer to a question about the relationship between Lipschitz retracts and the bounded approximation property in Banach spaces.

Findings

Necessary condition for Lipschitz retracts involves the bounded approximation property.

Addresses a question raised by Godefroy and Ozawa.

Links geometric properties of Banach spaces to approximation properties.

Abstract

In the present paper we prove that a necessary condition for a Banach space $X$ to admit a generating compact Lipschitz retract $K$, which satisfies an additional mild assumption on its shape, is that $X$ enjoys the Bounded Approximation Property. This is a partial solution to a question raised by Godefroy and Ozawa.