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

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

Abstract

In this chapter, we first study inequalities satisfied by any bounded linear operator \(u: A \rightarrow Y\) 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 \(u: A\rightarrow B(H)\) 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.

Mathematics Subject Classification (2000):

  • primary 47AO5
  • 46LO5
  • 43A65 secondary 47A20
  • 47B 10
  • 42B30
  • 46E40
  • 46L57
  • 47C 15
  • 47L20

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Correspondence to Gilles Pisier .

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© 2001 Springer-Verlag Berlin/Heidelberg

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Pisier, G. (2001). 7. The similarity problem for cyclic homomorphisms on a C*-algebra. In: Similarity Problems and Completely Bounded Maps. Lecture Notes in Mathematics, vol 1618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44563-0_8

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  • DOI: https://doi.org/10.1007/978-3-540-44563-0_8

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41524-4

  • Online ISBN: 978-3-540-44563-0

  • eBook Packages: Springer Book Archive