We combine the notion of norming algebra introduced by Pop, Sinclair and
Smith with a result of Pisier to show that if A_1 and A_2 are operator
algebras, then any bounded epimorphism of A_1 onto A_2 is completely bounded
provided that 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 A_i of C*-diagonals (C_i,D_i) (i=1,2)
satisfying D_i \subseteq A_i \subseteq C_i extends uniquely to a *-isomorphism
of the C*-algebras generated by A_1 and A_2; this generalizes results of
Muhly-Qiu-Solel and Donsig-Pitts.Comment: 9 page
