Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

Abstract

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 A-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 C A i C C, 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.