Remarks on the similarity degree of an operator algebra

Abstract

The "similarity" degree of a unital operator algebra $A$ was defined and studied in two recent papers of ours, where in particular we showed that it coincides with the "length" of an operator algebra. This paper brings several complements: we give direct proofs (with slight improvements) of several known facts on the length which were only known via the degree, and we show that the length of a type $I{I}_1$ factor with property $\Gamma$ is at most 5, improving on a previous bound ($\le 44$) due to E. Christensen.