A property in vector-valued function spaces

TL;DR

This paper investigates a geometric property in vector-valued function spaces that characterizes when these spaces have the Mazur-Ulam property, establishing equivalences across various function space constructions.

Contribution

It introduces a property equivalent to generalized-lushness for separable spaces and proves its preservation across different vector-valued function spaces.

Findings

The property is equivalent to generalized-lushness for separable spaces.

The property holds for $C(K,X)$ if and only if it holds for $X$.

The property also holds for $L_(,X)$ and $L_(,X)$ under similar conditions.

Abstract

This paper deals with a property which is equivalent to generalised-lushness for separable spaces. It thus may be seemed as a geometrical property of a Banach space which ensures the space to have the Mazur-Ulam property. We prove that if a Banach space $X$ enjoys this property if and only if $C(K,X)$ enjoys this property. We also show the same result holds for $L_\infty(\mu,X)$ and $L_1(\mu,X)$.