The similarity problem for cyclic homomorphisms on a C*-algebra

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


In this chapter, we first study inequalities satisfied by any bounded linear operator u: A→Y on a C*-algebra with values in a Banach space Y. The case when Y is another C*-algebra is of particular interest. Then we turn to homomorphisms u: A →* B(H) and prove that, if u is cyclic (= has a cyclic vector), boundedness implies complete boundedness. Hence bounded cyclic homomorphisms are similar to *-representations. This extends to homomorphisms with finite cyclic sets. We also include the positive solution to the similarity problem for C*-algebras without tracial states and for nuclear C*-algebras. Finally, we show that for a given C*-algebra, the similarity problem and the derivation problem are equivalent.


Unitary Representation Finite Sequence Isometric Embedding Tracial State Cyclic Vector 
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Notes and Remarks on Chapter 7

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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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