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

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.