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Geometric relations between spaces of nuclear operators and spaces of compact operators


Authors: Elói Medina Galego and Ronald Paternina Salguedo
Journal: Proc. Amer. Math. Soc. 140 (2012), 1643-1658
MSC (2010): Primary 46B03, 46B25; Secondary 47B10
Published electronically: August 22, 2011
MathSciNet review: 2869149
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Abstract: We extend and provide a vector-valued version of some results of C. Samuel about the geometric relations between the spaces of nuclear operators $ {\mathcal N}(E, F)$ and spaces of compact operators $ {\mathcal K}(E, F)$, where $ E$ and $ F$ are Banach spaces $ C(K)$ of all continuous functions defined on the countable compact metric spaces $ K$ equipped with the supremum norm.

First we continue Samuel's work by proving that $ {\mathcal N} (C(K_{1}), C(K_{2}))$ contains no subspace isomorphic to $ {\mathcal K} (C(K_{3}), C(K_{4}))$ whenever $ K_1$, $ K_{2}$, $ K_{3}$ and $ K_{4}$ are arbitrary infinite countable compact metric spaces.

Then we show that it is relatively consistent with ZFC that the above result and the main results of Samuel can be extended to $ C(K_{1}, X)$, $ C(K_{2}, Y)$, $ C(K_{3}, X)$ and $ C(K_{4}, Y)$ spaces, where $ K_1$, $ K_{2}$, $ K_{3}$ and $ K_{4}$ are arbitrary infinite totally ordered compact spaces; $ X$ comprises certain Banach spaces such that $ X^*$ are isomorphic to subspaces of $ l_1$; and $ Y$ comprises arbitrary subspaces of $ l_p$, with $ 1<p< \infty$.

Our results cover the cases of some non-classical Banach spaces $ X$ constructed by Alspach, by Alspach and Benyamini, by Benyamini and Lindenstrauss, by Bourgain and Delbaen and also by Argyros and Haydon.


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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

Ronald Paternina Salguedo
Affiliation: Department of Mathematics, University of São Paulo, São Paulo, Brazil 05508-090
Email: ronaldep@ime.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11006-2
Keywords: Isomorphic classifications of spaces of compact operators and spaces of nuclear operators
Received by editor(s): December 15, 2010
Received by editor(s) in revised form: January 12, 2011
Published electronically: August 22, 2011
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.