Lie ideals in properly infinite C*-algebras

TL;DR

This paper characterizes the structure of Lie ideals in properly infinite C*-algebras and related algebras, showing they correspond uniquely to two-sided ideals, thus clarifying their algebraic organization.

Contribution

It establishes a one-to-one correspondence between Lie ideals and two-sided ideals in properly infinite C*-algebras, answering a longstanding question and extending to real rank zero C*-algebras and von Neumann algebras.

Findings

Lie ideals in properly infinite C*-algebras are uniquely associated with two-sided ideals.

The structure of Lie ideals is fully determined by the lattice of two-sided ideals.

Results solve open problems in the theory of operator 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 Bre\v{s}ar, Kissin, and Shulman.