A Note on The Mazur-Ulam Property of Almost-CL-spaces

TL;DR

This paper introduces the (T)-property and demonstrates that Banach spaces with this property, including certain almost-CL-spaces and spaces of continuous functions, possess the Mazur-Ulam property, expanding understanding of geometric properties in functional analysis.

Contribution

The paper defines the (T)-property and proves that Banach spaces with this property have the Mazur-Ulam property, including specific classes like almost-CL-spaces and vector-valued continuous function spaces.

Findings

Almost-CL-spaces with smooth points have the MUP.

Two-dimensional spaces with hexagonal unit spheres have the MUP.

The MUP is stable under $c_0$- and $ ext{l}_1$-sums.

Abstract

We introduce the (T)-property, and prove that every Banach space with the (T)-property has the Mazur-Ulam property (briefly MUP). As its immediate applications, we obtain that almost-CL-spaces admitting a smooth point(specially, separable almost-CL-spaces) and a two-dimensional space whose unit sphere is a hexagon has the MUP. Furthermore, we discuss the stability of the spaces having the MUP by the $c_0$- and $\ell_1$-sums, and show that the space $C(K,X)$ of the vector-valued continuous functions has the the MUP, where $X$ is a separable almost-CL-space and $K$ is a compact metric space.