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3. Completely bounded maps

Part of the Lecture Notes in Mathematics book series (LNM,volume 1618)

Abstract

In this chapter, we first prove a fundamental criterion for an operator between Banach spaces to factor through a Hilbert space. Then we turn to the notion of complete boundedness (which is crucial for these notes). We prove a fundamental factorization/extension theorem for completely bounded maps, and give several consequences. In this viewpoint, the underlying idea is the same in both cases (completely bounded maps or operators factoring through Hilbert space). At the end of this chapter, we give several examples of bounded linear maps which are not completely bounded, and related norm estimates.

Mathematics Subject Classification (2000):

  • primary 47AO5
  • 46LO5
  • 43A65 secondary 47A20
  • 47B 10
  • 42B30
  • 46E40
  • 46L57
  • 47C 15
  • 47L20

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Correspondence to Gilles Pisier .

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© 2001 Springer-Verlag Berlin/Heidelberg

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Pisier, G. (2001). 3. Completely bounded maps. In: Similarity Problems and Completely Bounded Maps. Lecture Notes in Mathematics, vol 1618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44563-0_4

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  • DOI: https://doi.org/10.1007/978-3-540-44563-0_4

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41524-4

  • Online ISBN: 978-3-540-44563-0

  • eBook Packages: Springer Book Archive