Lie ideals in properly infinite $C^*$-algebras
Abstract
We show that every Lie ideal in a unital, properly infinite C*-algebra is commutator equivalent to a unique two-sided ideal. It follows that the Lie ideal structure of such a C*-algebra is concisely encoded by its lattice of two-sided ideals. This answers a question of Robert in this setting. We obtain similar structure results for Lie ideals in unital, real rank zero C*-algebras without characters. As an application, we show that every Lie ideal in a von Neumann algebra is related to a unique two-sided ideal, which solves a problem of Brešar, Kissin, and Shulman.
Fields of Study & Subjects
Advanced Operator Algebra Research
Advanced Topics in Algebra
Algebraic structures and combinatorial models
Publication details
- Journal / source
- Publicacions Matemàtiques
- Volume
- 70
- Pages
- 525–548
- Publication date
- 17 June 2026
- Language
- EN