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A complete classification of the spaces of compact operators on $ C([1,\alpha], l_{p})$ spaces, $ 1<p< \infty$


Authors: Dale E. Alspach and Elói Medina Galego
Journal: Proc. Amer. Math. Soc. 143 (2015), 2495-2506
MSC (2010): Primary 46B03; Secondary 46B25
DOI: https://doi.org/10.1090/S0002-9939-2015-12441-0
Published electronically: February 3, 2015
MathSciNet review: 3326031
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Abstract: We complete the classification, up to isomorphism, of the spaces of compact operators on $ C([1, \gamma ], l_{p})$ spaces, $ 1<p< \infty $. In order to do this, we classify, up to isomorphism, the spaces of compact operators $ {\mathcal K}(E, F)$, where $ E= C([1, \lambda ], l_{p})$ and $ F=C([1, \xi ], l_q)$ for arbitrary ordinals $ \lambda $ and $ \xi $ and $ 1< p \leq q< \infty $.

More precisely, we prove that it is relatively consistent with ZFC that for any infinite ordinals $ \lambda $, $ \mu $, $ \xi $ and $ \eta $ the following statements are equivalent:

(a)
$ {\mathcal K}(C([1, \lambda ], l_{p}), C([1, \xi ], l_{q})) $ is isomorphic to $ {\mathcal K}(C([1, \mu ], l_{p}), C([1, \eta ], l_{q})) .$

(b)
$ \lambda $ and $ \mu $ have the same cardinality and $ C([1,\xi ])$ is isomorphic to $ C([1, \eta ])$ or there exists an uncountable regular ordinal $ \alpha $ and $ 1 \leq m, n < \omega $ such that $ C([1, \xi ])$ is isomorphic to $ C([1, \alpha m])$ and $ C([1, \eta ])$ is isomorphic to $ C([1, \alpha n])$.

Moreover, in ZFC, if $ \lambda $ and $ \mu $ are finite ordinals and $ \xi $ and $ \eta $ are infinite ordinals, then the statements (a) and (b$ '$) are equivalent.

(b')
$ C([1,\xi ])$ is isomorphic to $ C([1, \eta ])$ or there exists an uncountable regular ordinal $ \alpha $ and $ 1 \leq m, n \le \omega $ such that $ C([1, \xi ])$ is isomorphic to $ C([1, \alpha m])$ and $ C([1, \eta ])$ is isomorphic to $ C([1, \alpha n])$.

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Additional Information

Dale E. Alspach
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: alspach@math.okstate.edu

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/S0002-9939-2015-12441-0
Keywords: $C([1, \alpha])$ spaces, $l_{p}$ spaces, spaces of compact operators on $C([1, \alpha], l_{p})$ spaces, isomorphic classifications.
Received by editor(s): August 12, 2013
Received by editor(s) in revised form: January 1, 2014, and January 8, 2014
Published electronically: February 3, 2015
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society

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