Skip to main content
SpringerLink
Log in

Completely bounded maps

  • Chapter
Similarity Problems and Completely Bounded Maps

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

  • 856 Accesses

Summary

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Haagerup U.: Decomposition of completely bounded maps on operator alge bras.Unpubli hed manuscript. Sept. 1980.

    Google Scholar 

  2. Paulsen V. Completely bounded maps and dilations. Pitman Research Notes in Math. 146, Longman, Wiley, New York, 1986.

    Google Scholar 

  3. Paulsen V. Completely bounded maps on C’-algebras and invariant operator ranges. Proc. Amer. Math. Soc. 86 (1982) 91 – 96.

    Google Scholar 

  4. Arveson W.: Subalgebras of C’-algebras. Acta Math. 123 (1969) 141–224. Part II. Acta Math. 128 (1972) 271 – 308.

    MathSciNet  MATH  Google Scholar 

  5. Wittstock G.: Ein operatorwertigen Hahn-Banach Satz. J. Funct. Anal. 40 (1981) 127 – 150.

    Article  MathSciNet  MATH  Google Scholar 

  6. Wittstock G.: On matrix order and convexity. Functional analysis: surveys and recent results. Math. Studies 90, p. 175-188. North Holland, Amsterdam, 1984.

    Google Scholar 

  7. Stinespring W.: Positive functions on C’-algebras. Proc. Amer. Math. Soc. 6 (1966) 211 – 216.

    Google Scholar 

  8. Lindenstrauss J. and Pelczynski A.: Absolutely summing operators in Lp spaces and their applications. Studia Math. 29 (1968) 275 – 326.

    MathSciNet  Google Scholar 

  9. Grothendieck A.: Résumé de la théorie métrique des produits tensoriels topo logiques. Boll. Soc. Mat. Sao-Paulo 8 (1956) 1 – 79.

    MathSciNet  Google Scholar 

  10. Pietsch A. Operator ideals. North-Holland Amsterdam 1978.

    Google Scholar 

  11. Kwapien S.: On operators factorizable through Lp-space. Bull. Soc. Math. France Mémoire 31–32 (1972) 215 – 225.

    MathSciNet  Google Scholar 

  12. Pisier G.: Completely bounded maps between sets of Banach space operators. Indiana Univ. Math. J. 39 (1990) 251 – 277.

    MathSciNet  Google Scholar 

  13. Smith R.: Completely bounded maps between C’-algebras. J. London Math. Soc. 27 (1983) 157 – 166.

    Article  MathSciNet  Google Scholar 

  14. Paulsen V. Representations of function algebras, abstract operator spaces and Banach space geometry. J. Funct. Anal. 109 (1992) 113 – 129.

    Article  MathSciNet  Google Scholar 

  15. Haagerup U.: Injectivity and decomposition of completely bounded maps. In “Operator algebras and their connection with topology and ergodic theory”, Springer Lecture Notes in Math. 1132 (1985) 91 – 116.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Texas A&M University, 77843, College Station, TX, USA

    Gilles Pisier

  2. Equipe d’Analyse, Université Paris VI, Case 186, 4 Place Jussieu, F-75252, Paris Cedex 05, France

    Gilles Pisier

Authors
  1. Gilles Pisier
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Pisier, G. (1996). Completely bounded maps. In: Similarity Problems and Completely Bounded Maps. Lecture Notes in Mathematics, vol 1618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21537-1_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-21537-1_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60322-1

  • Online ISBN: 978-3-662-21537-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation