Generalized-Lush Spaces and the Mazur-Ulam Property

2012/10/27 by Dongni Tan, Tan, Dongni, Xujian Huang +3
Mathematics · #46A22 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 46B04 #Secondary 46B20 #math.FA #msc:46A22 #msc:46B04 #msc:46B20

paper · pdf · doi:10.48550/arxiv.1210.7324

16 pages

arxiv created 2012/10/27 · arxiv updated 2012/10/30

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 c₀-, l₁- and l_∞-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.

Related

• Some remarks on generalised lush spaces
• A Note on The Mazur-Ulam Property of Almost-CL-spaces
• A property in vector-valued function spaces
• The Mazur-Ulam property for a Banach space which satisfies a separation condition
• Generalized-lush spaces revisited