In the last chapter we saw how the abstract characterization of operator algebras led to a number of factorization formulas for certain universal operator algebras. However, this theory was an isometric theory. In this chapter we focus on the isomorphic theory of operator algebras and applications to similarity questions.

We present Pisier's remarkable work on similarity degree and factorization degree, and Blecher's characterization of operator algebras up to cb isomorphism.

Pisier's work shows that for an operator algebra B, every bounded homomorphism is completely bounded if and only if the type of factorization occurring in the study of MAXA(B) can be carried out with uniform control on the number of factors needed. The least such integer is the factorization degree of the algebra.

Pisier's work has a number of deep implications in the study of bounded representations of groups and in the study of Kadison's similarity conjecture. We focus primarily on Kadison's conjecture, that every bounded homomorphism of a C*-algebra into B(H) is similar to a ∗-homomorphism. Thus, we will show that Kadison's conjecture is equivalent to the existence of an integer d such that every C*-algebra has factorization degree at most d.

A pivotal role in Pisier's work is played by the universal operator algebra of an operator space.