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On isomorphic classifications of spaces of compact operators


Author: Elói Medina Galego
Journal: Proc. Amer. Math. Soc. 137 (2009), 3335-3342
MSC (2000): Primary 46B03, 46B25; Secondary 47B10
DOI: https://doi.org/10.1090/S0002-9939-09-09828-1
Published electronically: May 13, 2009
MathSciNet review: 2515403
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Abstract: We prove an extension of the classical isomorphic classification of Banach spaces of continuous functions on ordinals. As a consequence, we give complete isomorphic classifications of some Banach spaces $ {\mathcal K}(X, Y^{\eta})$, $ \eta \geq \omega$, of compact operators from $ X$ to $ Y^{\eta}$, the space of all continuous $ Y$-valued functions defined in the interval of ordinals $ [1, \eta]$ and equipped with the supremum norm. In particular, under the Continuum Hypothesis, we extend a recent result of C. Samuel by classifying, up to isomorphism, the spaces $ {\mathcal K}(X^{\xi}, c_{0}(\Gamma)^{\eta})$, where $ \omega \leq \xi < \omega_1$, $ \eta \geq \omega$, $ \Gamma$ is a countable set, $ X$ contains no complemented copy of $ l_1$, $ X^*$ has the Mazur property and the density character of $ X^{**}$ is less than or equal to $ \aleph_{1}$.


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  • 1. C. Bessaga, A. Pełczyński, Spaces of continuous functions. IV, Studia Math. XIX (1960), 53-62. MR 0113132 (22:3971)
  • 2. L. Burlando, On subspaces of direct sums of infinite sequences of Banach spaces. Atti Accad. Ligure Sci. Lett. 46 (1989), 96-105. MR 1098789 (92b:46019)
  • 3. J. Diestel, J.J. Uhl, Jr., Vector Measures, Mathematical Surveys 15, Amer. Math. Soc., Providence, RI (1977). MR 0453964 (56:12216)
  • 4. G. A. Edgar, Measurability in a Banach space. II, Indiana Univ. Math. J. 28 (1979), 559-579. MR 542944 (81d:28016)
  • 5. E. M. Galego, How to generate new Banach spaces non-isomorphic to their Cartesian squares, Bull. Polish Acad. Sci. Math. 47 (1999), 1, 21-25. MR 1685684 (2001b:46015)
  • 6. E. M. Galego, Banach spaces of continuous vector-valued functions of ordinals, Proc. Edinb. Math. Soc. (2) 44 (2001), 1, 49-62. MR 1879208 (2002k:46064)
  • 7. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 1955 (1955), no. 16. MR 0075539 (17:763c)
  • 8. S. P. Gul'ko, A. V. Os'kin, Isomorphic classification of spaces of continuous functions on totally ordered bicompacta, Functional Anal. Appl. 9 (1975), 56-57. MR 0377489 (51:13661)
  • 9. T. Kappeler, Banach spaces with the condition of Mazur, Math. Z. 191 (1986), 623-631. MR 832820 (87h:46040)
  • 10. S. V. Kislyakov, Classification of spaces of continuous functions of ordinals, Siberian Math. J. 16 (1975), no. 2, 226-231.
  • 11. J. Lindenstrauss, L. Tzafriri, Classical Banach spaces. I. Sequence Spaces, Springer-Verlag, Berlin-New York (1977). MR 0500056 (58:17766)
  • 12. D. Leung, On Banach spaces with Mazur's property, Glasgow Math. J. 33 (1991), 51-54. MR 1089953 (92b:46021)
  • 13. H. P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13-36. MR 0270122 (42:5015)
  • 14. H. P. Rosenthal, On injective Banach spaces and the spaces $ L^{\infty}(\mu)$ for finite measures $ \mu$, Acta Math. 124 (1970), 205-248. MR 0257721 (41:2370)
  • 15. C. Samuel, Sur la reproductibilité des espaces $ l_p$, Math. Scand. 45 (1979), 103-117. MR 567436 (81e:46062)
  • 16. C. Samuel, Sur les sous-espaces de $ l_p {\hat{\hat\otimes}} l_q$. Math. Scand. 47 (1980), 247-250. MR 612698 (82i:46029)
  • 17. C. Samuel, On spaces of operators on $ C(Q)$ spaces ($ Q$ countable metric space), Proc. Amer. Math. Soc. 137 (2009), no. 3, 965-970. MR 2457436
  • 18. Z. Semadeni, Banach spaces non-isomorphic to their Cartesian squares. II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 81-84. MR 0115074 (22:5877)
  • 19. A. Wilansky, Mazur spaces, Internat. J. Math. Math. Sci. 4 (1981), 39-53. MR 606656 (82f:46001)

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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/S0002-9939-09-09828-1
Keywords: Isomorphic classifications of spaces of continuous functions, compact operators
Received by editor(s): August 28, 2008
Published electronically: May 13, 2009
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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