Pure C*-algebras

2024/06/16 by Ramon Antoine, Antoine, Ramon, Francesc Perera +5 · 3 citations
Computer Science · Mathematics · #46L05 (Primary) 19K14 #46L80 #46L85 (Secondary) #Advanced Algebra and Logic #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2406.11052

openalex publication_date 2024/06/16 · openalex created_date 2024/06/19 · openalex updated_date 2026/07/28

Abstract

We demonstrate that pure C*-algebras form a robust class by proving that pureness follows from very weak comparison and divisibility properties. Using this, we show that every simple, non-elementary C*-algebra with a unique quasitrace and with very mild comparison is pure, and, as a result, has strict comparison. Furthermore, sufficiently non-commutative C*-algebras of stable rank one and with weak comparison are likewise pure. We also show that adequately non-elementary C*-algebras with finite nuclear dimension are pure, which leads to the verification of the non-simple Toms-Winter conjecture for a large class of C*-algebras.

Cited by

• The Global Glimm Property for C*-algebras of topological dimension zero
• Prime and semiprime Lie ideals in C*-algebras
• Extensions of pure C*-algebras

Related

• Extensions of pure C*-algebras
• Tracial approximate divisibility and stable rank one
• On classification of non-unital simple amenable C*-algebras, I
• Nuclear dimension and Z-stability of pure C*-algebras
• The relative commutant of separable C*-algebras of real rank zero