Tracial oscillation zero and stable rank one

TL;DR

The paper investigates how tracial oscillation and its approximation zero variant govern structural regularity in separable simple C*-algebras with strict comparison. It establishes that tracial approximate oscillation zero is equivalent to stable rank one and to the surjectivity of the Gamma map on the Cuntz semigroup, unifying rank-theoretic and functional-analytic perspectives. It also shows that Gamma surjectivity, together with strict comparison, implies almost stable rank one actually yields stable rank one, and introduces a tracial matricial property (TM) that encapsulates a nice matricial structure compatible with these rank-theoretic conditions. Additional results on sequence algebras and range of dimension functions reinforce the connection between tracial data and global regularity properties, contributing to the broader program surrounding the Toms-Winter conjecture and Z-stability in non-nuclear settings. Overall, the work provides a robust link between tracial oscillations, Cuntz-semigroup data, and stable rank-type regularity in simple C*-algebras.

Abstract

Let $A$ be a separable (not necessarily unital) simple $C^*$-algebra with strict comparison. We show that if $A$ has tracial approximate oscillation zero then $A$ has stable rank one and the canonical map $Γ$ from the Cuntz semigroup of $A$ to the corresponding affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple $C^*$-algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map $Γ$ is surjective.

Theorems & Definitions (139)

• Theorem 1.1
• Definition 2.3
• Definition 2.4
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Proposition 2.9: Lemma 4.5 of eglnp
• proof
• Proposition 2.10
• proof