Completely bounded isomorphisms of operator algebras and similarity to complete isometries

Abstract: A well-known theorem of Paulsen says that if $\mathcal{A}$ is a unital operator algebra and $\phi:\mathcal{A}\to B(\mathcal{H})$ is a unital completely bounded homomorphism, then $\phi$ is similar to a completely contractive map $\phi'$. Motivated by classification problems for Hilbert space contractions, we are interested in making the inverse $\phi'^{-1}$ completely contractive as well whenever the map $\phi$ has a completely bounded inverse. We show that there exist invertible operators $X$ and $Y$ such that the map $$ XaX^{-1}\mapsto Y\phi(a)Y^{-1} $$ is completely contractive and is "almost" isometric on any given finite set of elements from $\mathcal{A}$ with non-zero spectrum. Although the map cannot be taken to be completely isometric in general, we show that this can be achieved if $\mathcal{A}$ is completely boundedly isomorphic to either a $C^*$-algebra or a uniform algebra. In the case of quotient algebras of $H^\infty$, we translate these conditions in function theoretic terms and relate them to the classical Carleson condition.