Generalized-Lush Spaces and the Mazur-Ulam Property

TL;DR

This paper introduces generalized-lush spaces (GL-spaces), expanding the class of Banach spaces known to have the Mazur-Ulam property, and explores their stability under common Banach space operations.

Contribution

It defines GL-spaces, shows their stability under sums, and proves the Mazur-Ulam property for a broader class called local-GL-spaces, including all lush spaces.

Findings

GL-spaces include almost-CL-spaces and separable lush spaces.

The space C(K,E) is a GL-space if E is.

GL-spaces are stable under c_0, l_1, and l_infinity sums.

Abstract

We introduce a new class of Banach spaces, called generalized-lush spaces (GL-spaces for short), which contains almost-CL-spaces, separable lush spaces (specially, separable $C$-rich subspaces of $C(K)$), and even the two-dimensional space with hexagonal norm. We obtain that the space $C(K,E)$ of the vector-valued continuous functions is a GL-space whenever $E$ is, and show that the GL-space is stable under $c_0$-, $l_1$- and $l_\infty$-sums. As an application, we prove that the Mazur-Ulam property holds for a larger class of Banach spaces, called local-GL-spaces, including all lush spaces and GL-spaces. Furthermore, we generalize the stability properties of GL-spaces to local-GL-spaces. From this, we can obtain many examples of Banach spaces having the Mazur-Ulam property.