Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if $\mathcal{A}_1$ and $\mathcal{A}_2$ are operator algebras, then any bounded epimorphism of $\mathcal{A}_1$ onto $\mathcal{A}_2$ is completely bounded provided that $\mathcal{A}_2$ contains a norming $C^*$-subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a $C^*$-algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a $C^*$-algebra is similar to a $*$-representation precisely when the image operator algebra $\lambda$-norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras $\mathcal{A}_i$ of $C^*$-diagonals $(\mathcal{C}_i, \mathcal{D}_i)$ ($i=1,2$) satisfying $\mathcal{D}_i\subseteq\mathcal{A}_i\subseteq \mathcal{C}_i$ extends uniquely to a $*$-isomorphism of the $\mathcal{C}^*$-algebras generated by $\mathcal{A}_1$ and $\mathcal{A}_2$; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.