The Similarity Degree of Some C*-algebras

We define the class of weakly approximately divisible unital C*-algebras and show that this class is closed under direct sums, direct limits, any tensor product with any C*-algebra, and quotients. A nuclear C*-algebra is weakly approximately divisible if and only if it has no finite-dimensional representations. We also show that Pisier's similarity degree of a weakly approximately divisible C*-algebra is at most 5

Free Orbit Dimension of Finite von Neumann Algebras

In this paper we introduce the concept of the upper free orbit-dimension of a finite von Neumann algebra, and we derive some of its basic properties. Using this concept, we are able to improve most of the applications of free entropy to finite von Neumann algebras, including those with Cartan subalgebras or simple masas, nonprime factors, those with property $T,$ those with property $\Gamma$, and thin factors. We provide more examples of II$_{1}$ von Neumann algebras whose Voiculesu's free entropy dimension is at most $1$. Our approach gives a strikingly easy way to get to the main results on free entropy dimension, and the paper is nearly self-contained.Comment: Revise