Some remarks on generalised lush spaces

Jan-David Hardtke¹

¹ Department of Mathematics Freie Universität Berlin Arnimallee 6 14195 Berlin, Germany

Studia Mathematica, Tome 231 (2015) no. 1, pp. 29-44

Résumé

D. Tan, X. Huang and R. Liu [Studia Math. 219 (2013)] recently introduced the notion of generalised lush (GL) spaces, which, at least for separable spaces, is a generalisation of the concept of lushness introduced by Boyko et al. [Math. Proc. Cambridge Philos. Soc. 142 (2007)]. The main result of D. Tan et al. is that every GL-space has the so called Mazur–Ulam property (MUP). In this note, we prove some further properties of GL-spaces, for example, every $M$-ideal in a GL-space is again a GL-space, ultraproducts of GL-spaces are again GL-spaces, and if the bidual $X^{**}$ of a Banach space $X$ is GL, then $X$ itself has the MUP.

Détail

DOI : 10.4064/sm8192-1-2016

Keywords: tan huang liu studia math recently introduced notion generalised lush spaces which least separable spaces generalisation concept lushness introduced boyko math proc cambridge philos soc main result tan every gl space has called mazur ulam property mup note prove further properties gl spaces example every m ideal gl space again gl space ultraproducts gl spaces again gl spaces bidual ** banach space itself has mup