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STRUCTURE TOPOLOGY AND EXTREME OPERATORS

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  19 October 2016

ANA M. CABRERA-SERRANO  and
JUAN F. MENA-JURADO
ANA M. CABRERA-SERRANO
Affiliation:
Universidad de Granada, Facultad de Ciencias, Departamento de Análisis Matemático, 18071 Granada, Spain email anich7@correo.ugr.es
JUAN F. MENA-JURADO*
Affiliation:
Universidad de Granada, Facultad de Ciencias, Departamento de Análisis Matemático, 18071 Granada, Spain email jfmena@ugr.es
*
✉jfmena@ugr.es
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Abstract

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We say that a Banach space $X$ is ‘nice’ if every extreme operator from any Banach space into $X$ is a nice operator (that is, its adjoint preserves extreme points). We prove that if $X$ is a nice almost $CL$-space, then $X$ is isometrically isomorphic to $c_{0}(I)$ for some set $I$. We also show that if $X$ is a nice Banach space whose closed unit ball has the Krein–Milman property, then $X$ is $l_{\infty }^{n}$ for some $n\in \mathbb{N}$. The proof of our results relies on the structure topology.

Keywords

extreme operatorstructure topologyalmost CL-spaceBanach space

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B04: Isometric theory of Banach spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 2 , April 2017 , pp. 315 - 321
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

Supported by Spanish MINECO and FEDER projects Nos MTM2012-31755 and MTM2015-65020-P and by Junta de Andalucía and FEDER Grant FQM-185.

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