Completely bounded homomorphisms and derivations

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


In this chapter, we study completely bounded homomorphisms u: A → B(H) when A ⊂ B(ℋ) is a subalgebra. We first consider the case when H and ℋ are Banach spaces but mostly concentrate on the Hilbert space case. In the latter case, we prove the fundamental result that a unital homomorphism is completely bounded if it is similar to a completely contractive one. Let δ: A → B(H) be a derivation on a C*-algebra. We show that δ is completely bounded if it is inner. When A is the disc algebra, we prove that an operator T on H is similar to a contraction iff it is completely polynomially bounded, or in other words if the associated homomorphism ff(T) is completely bounded. We discuss a variant for operators on a Banach space and give several related facts. Finally, we give examples showing that a bounded (and actually contractive) unital homomorphism on a uniform algebra is not necessarily completely bounded.


Convolution Suffix 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes and Remarks on Chapter 4

  1. [Sa]
    Sarason D.: Generalized interpolation in H°°. Trans. Amer. Math. Soc. 127 (1967) 179–203.MathSciNetMATHGoogle Scholar
  2. [P4]
    Pisier G.: Completely bounded maps between sets of Banach space operators. Indiana Univ. Math. J. 39 (1990) 251–277.MathSciNetGoogle Scholar
  3. [Had]
    Hadwin D.: Dilations and Hahn decompositions for linear maps. Canad. J. Math. 33 (1981) 826–839.MathSciNetMATHCrossRefGoogle Scholar
  4. [Wi1]
    Wittstock G.: Ein operatorwertigen Hahn-Banach Satz. J. Funct. Anal. 40 (1981) 127–150.MathSciNetMATHCrossRefGoogle Scholar
  5. [H1]
    Haagerup U.: Solution of the similarity problem for cyclic representations of C’-algebras. Annals of Math. 118 (1983) 215–240.MathSciNetMATHCrossRefGoogle Scholar
  6. [Pa1]
    Paulsen V. Completely bounded maps and dilations. Pitman Research Notes in Math. 146, Longman, Wiley, New York, 1986.Google Scholar
  7. [C2]
    Christensen E.: Extensions of derivations II. Math. Scand. 50 (1982) 111–122.MathSciNetMATHGoogle Scholar
  8. [J]
    Johnson B. E.: Cohomology in Banach algebras. Memoirs Amer. Math. Soc. 127 (1972).Google Scholar
  9. [Ri]
    Ringrose J.: Cohomology theory for operator algebras. Proc. Symp. Pure Math. 38 (1982) 229–252.MathSciNetCrossRefGoogle Scholar
  10. [K4]
    Kadison R.: Derivations of operator algebras. Ann. Math. 83 (1966) 280–293.MathSciNetMATHCrossRefGoogle Scholar
  11. [S]
    Sakai S.: Derivations of W’-algebras. Ann. Math. 83 (1966) 273–279.MATHCrossRefGoogle Scholar
  12. [SS]
    Sinclair A. and Smith R.: Hochschild cohomology of von Neumann algebras. LMS Lecture notes series. Cambridge Univ. Press, 1994.Google Scholar
  13. [Di2]
    Dixmier J.: les algèbres d’opérateurs dans l’espace Hilbertien (Algèbres de von Neumann). Gauthier-Villars, Paris, 1969. (Translated into: von Neumann algebras, North-Holland, 1981.)Google Scholar
  14. [C5]
    Christensen E.: Perturbation of operator algebras II. Indiana Math. J. 26 (1977), 891–904.MATHGoogle Scholar
  15. [Ar2]
    Arveson W.: Ten lectures on operator algebras. CBMS 55. Amer. Math. Soc. Providence (RI ) 1984.Google Scholar
  16. [St]
    Stampfli J.: The norm of a derivation. Pacific J. Math. 33 (1970) 737–747.MathSciNetMATHGoogle Scholar
  17. [Pa3]
    Paulsen V. Every completely polynomially bounded operator is similar to a contraction. J. Funct. Anal. 55 (1984) 1–17.MathSciNetMATHCrossRefGoogle Scholar
  18. [M]
    Mascioni V.: Ideals of the disc algebra, operators related to Hilbert space contractions, and complete boundedness. Houston J. Math. 20 (1994) 299–311.MathSciNetMATHGoogle Scholar
  19. [SN]
    Sz: Nagy B.: Completely continuous operators with uniformly bounded iterates. Publ. Math. Inst. Hungarian Acad. Sci. 4 (1959) 89–92.Google Scholar
  20. [Bo]
    Bourgain J.: On the similarity problem for polynomially bounded operators on Hilbert space. Israel J. Math. 54 (1986) 227–241.MathSciNetMATHGoogle Scholar
  21. [Ro]
    Rota G.C.: On models for linear operators. Comm. Pure Appl. Math. 13 (1960) 468–472.MathSciNetGoogle Scholar
  22. [Pet]
    Petrovic S.: A dilation Theory for Polynomially Bounded Operators. J. Funct. Anal. 108 (1992) 458–469.MathSciNetMATHCrossRefGoogle Scholar
  23. [PPP]
    Paulsen V., Pearcy C. and Petrovic S.: On centered and weakly centered operators. J. Funct. Anal. 128 (1995) 87–101.MathSciNetMATHCrossRefGoogle Scholar
  24. [Pe1]
    Peller V.: Estimates of functions of power bounded operators on Hilbert space. J. Oper. Theory 7 (1982) 341–372.MathSciNetMATHGoogle Scholar
  25. [Pa5]
    Paulsen V. Representations of function algebras, abstract operator spaces and Banach space geometry. J. Funct. Anal. 109 (1992) 113–129.MathSciNetCrossRefGoogle Scholar
  26. [DP]
    Douglas V. and Paulsen V.: Hilbert modules over function algebras. Pitman Longman 1989.Google Scholar
  27. [CCFW]
    Carlson J., Clark D., Foias C. and Williams J. Projective Hilbert A(D)modules. Preprint.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Gilles Pisier
    • 1
    • 2
  1. 1.Mathematics DepartmentTexas A&M UniversityCollege StationUSA
  2. 2.Equipe d’AnalyseUniversité Paris VIParis Cedex 05France

Personalised recommendations