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STRUCTURE TOPOLOGY AND EXTREME OPERATORS

  • ANA M. CABRERA-SERRANO (a1) and JUAN F. MENA-JURADO (a2)

Abstract

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.

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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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STRUCTURE TOPOLOGY AND EXTREME OPERATORS

  • ANA M. CABRERA-SERRANO (a1) and JUAN F. MENA-JURADO (a2)

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