A Modified Similarity Degree for C*-algebras

We define variants of Pisier’s similarity degree for unital C*-algebras and use direct integral theory to obtain new results. We prove that if every II${}_1$ factor representation of a separable C*-algebra $\mathcal{A}$ has property $\mathrm{\Gamma }$, then the similarity degree of $\mathcal{A}$ is at most 11.