Completely bounded homomorphisms of operator algebras

Abstract:

Let $A$ be a unital operator algebra. We prove that if $\rho$ is a completely bounded, unital homomorphism of $A$ into the algebra of bounded operators on a Hilbert space, then there exists a similarity $S$, with $\left \| {{S^{ - 1}}} \right \| \cdot \left \| S \right \| = {\left \| \rho \right \|_{cb}}$, such that ${S^{ - 1}}\rho ( \cdot )S$ is a completely contractive homomorphism. We also show how Rota’s theorem on operators similar to contractions and the result of Sz.-Nagy and Foias on the similarity of $\rho$-dilations to contractions can be deduced from this result.