The similarity degree of an operator algebra II

For every integer $d\ge 1$, there is a unital closed subalgebra $A_d\subset B(H)$withsimilaritydegreeequalpreciselyto$d$, in the sense of our previous paper. This means that for any unital homomorphism $u\colon A_d\to B(H)$wehave$|u|_{cb} \le K|u|^d$with$K>0$independentof$u$, 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 Thorbj\o rnsen. We also include several complements to our previous publications on the same subject.