4. Completely bounded homomorphisms and derivations

Abstract

In this chapter, we study completely bounded homomorphisms u: $A\rightarrow B(H)$ when $A \subset B(\mathcal{H})$ is a subalgebra. We first consider the case when H and $\mathcal{H}$ are Banach spaces but mostly concentrate on the Hilbert space case. In the latter case, we prove the Jundamental result that a unital homomorphism is completely bounded iff it is similar to a completely contractive one. Let $\delta: A \rightarrow B(H)$ be a derivation on a C^*-algebra. We show that $\delta$ is completely bounded iff it is inner. When A is the disc algebra, we prove that an operator T on H is similar to a contraction iff it is completely polynomially bounded, or in other words iff the associated homomorphism $f\rightarrow f (T)$ is completely bounded. We discuss a variant for operators on a Banach space and give several related facts. Finally, we give examples showing that a bounded (and actually contractive) unital homomorphism on a uniform algebra is not necessarily completely bounded.