The Similarity Degree of an Operator Algebra, Ii

For every integer d ≥ 1, there is a unital closed subalgebra A d ⊂ B(H) with similarity degree equal precisely to d, in the sense of our previous paper. This means that for any unital homomorphism u: A d → B(H) we have u cb ≤ Ku d with K > 0 independent of u, and the exponent d in this estimate cannot be improved. The proof that the degree is larger than d − 1 crucially uses an upper bound for the norms of certain Gaussian random matrices due to Haagerup and Thorbjørnsen. We also include several complements to our previous publications on the same subject. This article is a continuation of our earlier papers [P1, P2]. We denote by B(H) the algebra of all bounded operators on a Hilbert space H. Let A be a unital operator algebra i.e. a closed unital subalgebra of B(H). Assume that every bounded morphism (= unital homomorphism) u: A → B(H) is automatically completely bounded (c.b. in short). Then (cf. [P1]) there is an integer d and a constant K such that any such u satisfies u cb ≤ Ku d. The smallest d for which this holds is called the similarity degree of A and is denoted by d(A). By convention, we set d(A) = ∞ if there is a bounded morphism u: A → B(H) which is not c.b. The main result proved in section 2 below is Theorem 0.1. For any d ≥ 1, there is a (nonself-adjoint) unital operator algebra A d such that d(A d) = d. Note that the existence of unital (nonself-adjoint) operator algebras A with d(A) = ∞ is well known. In view of this, the preceding result is not too surprising. However its verification has proved to be much more difficult than expected, although the algebras A d themselves are rather canonical and easy to define. In [P1], we gave examples of C *-algebras with degree equal to 1, 2 and 3 but we could not construct any examples (self-adjoint or not) with finite degree > 3. The preceding result fills this gap in the nonself-adjoint case, but the case of C *-algebras remains open. Note that a well known conjecture of Kadison [Ka] implies, modulo [P1], that there is a universal bound for the similarity degree of C *-algebras, but we are convinced that the opposite is what happens. The proof …