Generalized-lush spaces and the Mazur–Ulam property

Dongni Tan¹ ; Xujian Huang¹ ; Rui Liu²

¹ Department of Mathematics Tianjin University of Technology 300384 Tianjin, China
² Department of Mathematics and LPMC Nankai University 300071 Tianjin, China

Studia Mathematica, Tome 219 (2013) no. 2, pp. 139-153

Résumé

We introduce a new class of Banach spaces, called generalized-lush spaces (GL-spaces for short), which contains almost-CL-spaces, separable lush spaces (in particular, separable $C$-rich subspaces of $C(K)$), and even the two-dimensional space with hexagonal norm. We find that the space $C(K,E)$ of vector-valued continuous functions is a GL-space whenever $E$ is, and show that the set of GL-spaces 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.

Détail

DOI : 10.4064/sm219-2-4

Keywords: introduce class banach spaces called generalized lush spaces gl spaces short which contains almost cl spaces separable lush spaces particular separable c rich subspaces even two dimensional space hexagonal norm space vector valued continuous functions gl space whenever set gl spaces stable under infty sums application prove mazur ulam property holds larger class banach spaces called local gl spaces including lush spaces gl spaces furthermore generalize stability properties gl spaces local gl spaces obtain many examples banach spaces having mazur ulam property