
Journal of Convex Analysis 27 (2020), No. 3, 979988 Copyright Heldermann Verlag 2020 On the Structure Topology on the Set of all Extreme Points of the Closed Unit Ball of the Dual of a Banach Space Ana M. CabreraSerrano Dep. de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Spain anich7@correo.ugr.es Juan F. MenaJurado Dep. de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Spain jfmena@ugr.es [Abstractpdf] Let $X$ be a real Banach space, and let $E_{X^*}$ stand for the set of all extreme points of the closed unit ball of $X^*$, endowed with the AlfsenEffros structure topology [see E.\,M.\,Alfsen and E.\,G.\,Effros, {\it Structure in real Banach spaces I, II}, Annals of Math. 96 (1972) 98128; ibid.\ 96 (1972) 12973]. The fact that, for a given $s^* \in E_{X^*}$, the set $\{\pm s^* \}$ is structurally open can be characterized in many apparently different ways, whenever $X$ is nice. (We recall that $X$ is said to be nice if every extreme operator from any Banach space to $X$ is a nice operator, i.e. its adjoint preserves extreme points.) As a consequence, we obtain new characterizations (as well as new proofs of known characterizations) of those nice Banach spaces which are isometrically isomorphic to $c_0(I)$ for some set $I$. Keywords: Banach space, extreme operator, nice operator, structure topology. MSC: 46B20, 46B04 [ Fulltextpdf (124 KB)] for subscribers only. 