Some remarks on generalised lush spaces

Abstract: X. Huang et al. 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 K. Boyko et al. in 2007. The main result of Huang et al. is that every GL-space has the so called Mazur-Ulam property (MUP). In this note, we will prove some properties of GL-spaces (further than those already established by Huang et al.), 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 still has the MUP.