The similarity degree of an operator algebra. II (Q1568199)

The author constructs for any integer \(d\geq 1\) a closed subalgebra \(A_d\subseteq B(H)\) with similarity degree equal to \(d\). This means that for any unital homomorphism \(u: A_d\to B(H)\) one has \(\|u\|_{cb}\leq K\|u\|^d\) where \(K\) is independent of \(u\) and \(\|\cdot\|_{cb}\) denotes the completely bounded norm. The existence of unital nonselfadjoint operator algebras \(A\) with \(d(A)=\infty\) was well-known. The proof that the operator algebras constructed in the paper are not of similarity degree \(d-1\) is actually quite involved and uses upper estimates of Haagerup and Thorbjørnsen on the norm of Gaussian random matrices with matrix coefficients. [For part I see Algebra Anal. 10, No. 1, 132-186 (1998; Zbl 0911.47038)].