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.


Hilbert Space Banach Space Operator Algebra Uniform Algebra Disc Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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

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