The similarity degree of an operator algebra, II

For every integer , there is a unital closed subalgebra with similarity degree equal precisely to d, in the sense of our previous paper. This means that for any unital homomorphism we have with independent of u, and the exponent d in this estimate cannot be improved. The proof that the degree is larger than crucially uses an upper bound for the norms of certain Gaussian random matrices due to Haagerup and Thorbj?rnsen. We also include several complements to our previous publications on the same subject. Received: 25 April, 1999; Accepted: 23 June, 1999