Prime and semiprime Lie ideals in C*-algebras

TL;DR

This paper characterizes semiprime and prime Lie ideals in C*-algebras using Dixmier ideals, establishing a clear correspondence with two-sided ideals and providing a comprehensive structural description.

Contribution

It introduces a novel description of semiprime and prime Lie ideals in C*-algebras via their associated two-sided ideals, expanding the understanding of their structure.

Findings

Semiprime Lie ideals correspond to semiprime two-sided ideals.

Prime Lie ideals correspond to prime two-sided ideals.

A Lie ideal is fully noncentral iff it is not contained in any prime Lie ideal.

Abstract

Using the theory of Dixmier ideals developed in previous work, we show that every semiprime Lie ideal in a C*-algebra arises as the full normalizer subspace of a semiprime two-sided ideal. This leads to a concise description of all semiprime Lie ideals in terms of semiprime two-sided ideals, and an analogous description of prime Lie ideals in terms of prime two-sided ideals. For unital C*-algebras without characters, we obtain a natural bijection between (semi)prime two-sided ideals and (semi)prime Lie ideals, and it follows that a Lie ideal is fully noncentral if and only if it is not contained in any prime Lie ideal.