Some remarks on generalised lush spaces

Abstract

X. Huang et al. recently introduced the notion of generalised lush (GL) spaces in [13], which, at least for separable spaces, is a generalisation of the concept of lushness introduced in [3]. The main result of [13] 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 in [13]), 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.

A property in vector-valued function spaces

“…Throughout what follows, we shall freely use without explicit mention an elementary fact that Definition 1.3 is equivalent to another one where the assumption: x 1 , x 2 ∈ S X is replaced by x 1 ∈ S X and x 2 ∈ B X . It should be remarked that the following observations were made in [9].…”

“…Moreover many stable properties for GL-spaces are established in [16], for example, it is established that the class of GL-spaces is stable under c 0 , l 1 and l ∞ -sums ([16, Theorem 2.11 and Proposition 2.12]) and that if X is a GL space then so is the space C(K, X) of all continuous functions from any compact Hausdorff space K into X ([16, Theorem 2.10]). Later Jan-David Hardtke in [9] stated that a large class of GL-spaces is stable under ultraproducts and under passing to a large class of F -ideals, in particular to M-ideals. And more, he introduced in [9] (with the help of an anonymous referee as is mentioned in the [10, 2.4 Lush spaces]) the following (at least formally) weaker version of GL-spaces: Definition 1.3.…”

“…Later Jan-David Hardtke in [9] stated that a large class of GL-spaces is stable under ultraproducts and under passing to a large class of F -ideals, in particular to M-ideals. And more, he introduced in [9] (with the help of an anonymous referee as is mentioned in the [10, 2.4 Lush spaces]) the following (at least formally) weaker version of GL-spaces: Definition 1.3. A Banach space X is said to have the property ( * * ) if for all x 1 , x 2 ∈ S X and every ε > 0, there exists a slice S x * := S(x * , ε) with x * ∈ S X * such that…”

Generalized-lush spaces revisited

“…Also it is shown that the class of GL-spaces is stable under c 0 -, 1 -and ∞ -sums and that the space C(K, X) is a GL-space whenever X is a GL-space, which gives a number of examples of spaces with the Mazur-Ulam property. In the same vein Jan-David Hardtke [5] demonstrated that the class of GL-spaces is stable under ultraproducts and under passing to a large class of F-ideals, in particular to M-ideals.…”

On Certain Geometric Properties in Banach Spaces of Vector-Valued Functions

“…This notion was introduced in the author's paper [17] (with the help of an anonymous referee) and the following observations were made: A test family for ( * * ) in X can be defined by…”