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Strict comparison and stable rank one

Huaxin Lin

Abstract

Let $A$ be a $σ$-unital finite simple $C^*$-algebra which has strict comparison property. We show that if the canonical map $Γ$ from the Cuntz semigroup to certain lower semi-continuous affine functions is surjective, then $A$ has tracial approximate oscillation zero and stable rank one. Equivalently, if $A$ has an almost unperforated and almost divisible Cuntz semigroup, then $A$ has stable rank one and tracial approximate oscillation zero.

Strict comparison and stable rank one

Abstract

Let be a -unital finite simple -algebra which has strict comparison property. We show that if the canonical map from the Cuntz semigroup to certain lower semi-continuous affine functions is surjective, then has tracial approximate oscillation zero and stable rank one. Equivalently, if has an almost unperforated and almost divisible Cuntz semigroup, then has stable rank one and tracial approximate oscillation zero.
Paper Structure (4 sections, 20 theorems, 101 equations)

This paper contains 4 sections, 20 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strict comparison
  4. Tracial approximate oscillation zero

Key Result

Theorem 1.1

Let $A$ be a $\sigma$-unital non-elementary simple $C^*$-algebra with strict comparison (which is not purely infinite). Then the following are equivalent: (1) The canonical map $\Gamma: \mathrm{Cu}(A)\to {\rm LAff}_+(\widetilde{QT}(A))$ is surjective. (2) $A$ has tracial approximate oscillation zer

Theorems & Definitions (47)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 37 more