The similarity problem for cyclic homomorphisms on a C*-algebra
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.
KeywordsUnitary Representation Finite Sequence Isometric Embedding Tracial State Cyclic Vector
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Notes and Remarks on Chapter 7
- [Ki]Kirchberg E.: The derivation and the similarity problem are equivalent. Preprint. August 94.Google Scholar