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Some remarks on generalised lush spaces

Volume 231 / 2015

Jan-David Hardtke Studia Mathematica 231 (2015), 29-44 MSC: Primary 46B20. DOI: 10.4064/sm8192-1-2016 Published online: 9 February 2016

Abstract

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.

Authors

  • Jan-David HardtkeDepartment of Mathematics
    Freie Universität Berlin
    Arnimallee 6
    14195 Berlin, Germany
    e-mail

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