LIE IDEALS IN PROPERLY INFINITE C∗-ALGEBRAS

Journal article, 2026

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. 2020 Mathematics Subject Classification: Primary: 46L05, 46L10. Secondary: 16W10, 17B60, 47B47.