The weak* similarity property on dual operator algebras

Abstract

This paper is devoted to dual operator algebras, that isw*-closed algebras of bounded operators on Hilbert space. We investigate unital dual operator algebrasA with the following weak* similarity property: for every Hilbert spaceH, anyw*-continuous unital homomorphism fromA intoB(H) is completely bounded and thus similar to a contractive one. We develop a notion of dual similarity degree for these algebras, in analogy with some recent work of Pisier on the similarity problem on operator algebras.