The Similarity Degree of Some C*-algebras

TL;DR

This paper introduces weakly approximately divisible C*-algebras, explores their properties, and establishes a bound on Pisier's similarity degree for this class, linking divisibility to representation theory.

Contribution

It defines a new class of C*-algebras, analyzes their closure properties, and relates their similarity degree to the absence of finite-dimensional representations.

Findings

Weakly approximately divisible C*-algebras are closed under key operations.

A nuclear C*-algebra is weakly approximately divisible iff it has no finite-dimensional representations.

Pisier's similarity degree for these algebras is at most 5.

Abstract

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.