A Modified Similarity Degree for C*-algebras

TL;DR

This paper introduces modified similarity degrees for unital C*-algebras, using direct integral theory to establish bounds based on properties of II$_{1}$ factor representations.

Contribution

It defines new variants of Pisier's similarity degree and proves a bound of 11 when all II$_{1}$ factor representations have property $ extGamma$.

Findings

If all II$_{1}$ factor representations of a separable C*-algebra have property $ extGamma$, then its similarity degree is at most 11.

Uses direct integral theory to derive new results on similarity degrees.

Provides a link between representation properties and similarity degree bounds.

Abstract

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 $\Gamma$, then the similarity degree of $\mathcal{A}$ is at most 11.