Abstract

Let XX be a real Banach space, and let EX∗E_{X^*} stand for the set of all extreme points of the closed unit ball of X∗X^*, endowed with the Alfsen-Effros structure topology [see E. M. Alfsen and E. G. Effros, Structure in real Banach spaces I, II, Annals of Math. 96 (1972) 98–128; ibid. 96 (1972) 129–73]. The fact that, for a given s∗∈EX∗s^* \in E_{X^*}, the set {±s∗}\{\pm s^* \} is structurally open can be characterized in many apparently different ways, whenever XX is nice. (We recall that XX is said to be nice if every extreme operator from any Banach space to XX 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 c0(I)c_0(I) for some set II.

Author information

Contact details are reproduced from the original publication and may be historical.

Ana M. Cabrera-Serrano

Dep. de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Spain

anich7@correo.ugr.es

Juan F. Mena-Jurado

Dep. de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Spain

jfmena@ugr.es

Keywords

  • Banach space
  • extreme operator
  • nice operator
  • structure topology

Mathematics Subject Classification

46B20, 46B04

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Suggested citation

A. M. Cabrera-Serrano, J. F. Mena-Jurado. “On the Structure Topology on the Set of all Extreme Points of the Closed Unit Ball of the Dual of a Banach Space.” Journal of Convex Analysis 27 (2020), No. 3, 979–988. https://doi.org/10.68381/jca27050

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