Similarity degree of a class of C$^*$-algebras

TL;DR

This paper investigates the similarity degree of certain C$^*$-algebras and von Neumann algebras, establishing bounds based on properties like Property $ ext{ extGamma}$ and Property c$^*$-$ ext{ extGamma}$, with implications for $ ext{ extZ}$-stability.

Contribution

It provides new bounds and exact values for the similarity degree of specific classes of C$^*$-algebras and von Neumann algebras based on their properties.

Findings

If $ ext{ extM}$ has Property $ ext{ extGamma}$, then its similarity degree is ≤ 5.

If $ ext{ extA}$ has Property c$^*$-$ ext{ extGamma}$, then its similarity degree is exactly 3.

A $ ext{ extZ}$-stable, separable, non-nuclear, unital C$^*$-algebra has similarity degree 3.

Abstract

Suppose that $\mathcal M$ is a countably decomposable type II$_1$ von Neumann algebra and $\mathcal A$ is a separable, non-nuclear, unital C$^*$-algebra. We show that, if $\mathcal M$ has Property $\Gamma$, then the similarity degree of $\mathcal M$ is less than or equal to $5$. If $\mathcal A$ has Property c$^*$-$\Gamma$, then the similarity degree of $\mathcal A$ is equal to $3$. In particular, the similarity degree of a $\mathcal Z$-stable, separable, non-nuclear, unital C$^*$-algebra is equal to $3$.