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Complete isomorphic classifications of some spaces of compact operators


Author: Elói Medina Galego
Journal: Proc. Amer. Math. Soc. 138 (2010), 725-736
MSC (2000): Primary 46B03, 46B25; Secondary 47B10
DOI: https://doi.org/10.1090/S0002-9939-09-10117-X
Published electronically: October 9, 2009
MathSciNet review: 2557189
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Abstract:

This paper is a continuation and a complement of our previous work on isomorphic classification of some spaces of compact operators. We improve the main result concerning extensions of the classical isomorphic classification of the Banach spaces of continuous functions on ordinals. As an application, fixing an ordinal $ \alpha$ and denoting by $ X^{\xi}$, $ \omega_{\alpha} \leq \xi < \omega_{\alpha+1}$, the Banach space of all $ X$-valued continuous functions defined in the interval of ordinals $ [0, \xi]$ and equipped with the supremum, we provide complete isomorphic classifications of some Banach spaces $ {\mathcal K}(X^{\xi}, Y^\eta)$ of compact operators from $ X^\xi$ to $ Y^\eta$, $ \eta \geq \omega$.

It is relatively consistent with ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that these results include the following cases:

1. $ X^{*}$ contains no copy of $ c_{0}$ and has the Mazur property, and $ Y= c_{0}(J)$ for every set $ J$.

2. $ X=c_{0}(I)$ and $ Y=l_{q}(J)$ for any infinite sets $ I$ and $ J$ and $ 1 \leq q < \infty$.

3. $ X=l_{p}(I)$ and $ Y=l_{q}(J)$ for any infinite sets $ I$ and $ J$ and $ 1 \leq q<p < \infty$.


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Additional Information

Elói Medina Galego
Affiliation: Department of Mathematics, University of São Paulo, São Paulo, Brazil 05508-090
Email: eloi@ime.usp.br

DOI: https://doi.org/10.1090/S0002-9939-09-10117-X
Keywords: Isomorphic classifications of spaces of continuous functions, compact operators
Received by editor(s): March 5, 2009
Received by editor(s) in revised form: July 10, 2009
Published electronically: October 9, 2009
Additional Notes: The author would like to thank the referee for several helpful comments and suggestions which have been incorporated into the current version of the paper
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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