Ideals of a C*-algebra generated by an operator algebra

In this paper, we consider ideals of a C*-algebra C*(B) generated by an operator algebra B. A closed ideal J ⊆ C*(B) is called a K-boundary ideal if the restriction of the quotient map on B has a completely bounded inverse with cb-norm equal to K^-1. For K = 1 one gets the notion of boundary ideals introduced by Arveson. We study properties of the K-boundary ideals and characterize them in the case when operator algebra λ-norms itself. Several reformulations of the Kadison similarity problem are given. In particular, the affirmative answer to this problem is equivalent to the statement that every bounded homomorphism from C*(B) onto B which is a projection on B is completely bounded. Moreover, we prove that Kadison's similarity problem is decided on one particular C*-algebra which is a completion of the *-double of M₂(ℂ).