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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© 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
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