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The Kalton centralizer on $ L_p[0,1]$ is not strictly singular

Author: Jesús Suárez de la Fuente
Journal: Proc. Amer. Math. Soc. 141 (2013), 3447-3451
MSC (2010): Primary 46B20, 46B07, 46A16
Published electronically: June 5, 2013
MathSciNet review: 3080167
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Abstract: We prove that the Kalton centralizer on $ L_p[0,1]$, for $ 0<p<\infty $, is not strictly singular: in all cases there is a Hilbert subspace on which it is trivial. Moreover, for $ 0<p<2$ there are copies of $ \ell _q$, with $ p<q<2$, on which it becomes trivial. This is in contrast to the situation for $ \ell _p$ spaces, in which the Kalton-Peck centralizer is strictly singular.

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  • 1. F. Albiac, N. J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, 233. Springer-Verlag. MR 2192298 (2006h:46005)
  • 2. F. Cabello, J. M. F. Castillo, J. Suárez, On strictly singular non-linear centralizers, Nonlinear Anal. 75 (2012), no. 7, 3313-3321.
  • 3. J. M. F. Castillo, M. González, Three-space problems in Banach space theory, Lecture Notes in Mathematics, 1667. Springer-Verlag, Berlin, 1997. MR 1482801 (99a:46034)
  • 4. J. M. F. Castillo, Y. Moreno, Singular and cosingular exact sequences of quasi-Banach spaces, Arch. Math. 88 (2007), No. 2, 123-132. MR 2299035 (2008b:46004)
  • 5. N. J. Kalton, Differentials of complex interpolation processes for Köthe function spaces, Trans. Amer. Math. Soc. 333 (1992), No. 2, 479-529. MR 1081938 (92m:46111)
  • 6. N. J. Kalton, Nonlinear commutators in interpolation theory, Mem. Amer. Math. Soc. 73 (1988), No. 385. MR 938889 (89h:47104)
  • 7. N. J. Kalton, N. T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1-30. MR 542869 (82g:46021)
  • 8. M. Ledoux and M. Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 23. Springer-Verlag. MR 2814399
  • 9. V. D. Milman and G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Lecture Notes in Mathematics, 1200. Springer-Verlag, Berlin, 1986. MR 856576 (87m:46038)
  • 10. R. L. Schilling, Measures, integrals and martingales. Cambridge University Press, New York, 2005. MR 2200059 (2007g:28001)
  • 11. P. Wojtaszczyk, Banach spaces for analysts. Cambridge Studies in Advances Mathematics, 25. Cambridge University Press, Cambridge, 1991. MR 1144277 (93d:46001)

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Additional Information

Jesús Suárez de la Fuente
Affiliation: Escuela Politécnica, Universidad de Extremadura, Avenida Universidad s/n, 10071 Cáceres, Spain

Received by editor(s): March 31, 2011
Received by editor(s) in revised form: October 20, 2011, October 24, 2011, November 16, 2011, and December 8, 2011
Published electronically: June 5, 2013
Additional Notes: The author was partially supported by MTM2010-20190-C02-01 and Junta de Extremadura CR10113 “IV Plan Regional I+D+i”, Ayudas a Grupos de Investigación
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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