Completely bounded homomorphisms and derivations

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

Summary

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.

Keywords

Convolution Suffix 

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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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