Prime and semiprime Lie ideals in C*-algebras

2025/06/13 by Eusebio Gardella, Gardella, Eusebio, Kan Kitamura +3 · 1 citation
Mathematics · #16W10 #17B60 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #Primary 46L05. Secondary 16N60 #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2506.11819

openalex publication_date 2025/06/13 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

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.

Citations

• Fully noncentral Lie ideals and invariant additive subgroups in rings
• Pure C*-algebras
• Semiprime ideals in C*-algebras
• Prime ideals in C*-algebras and applications to Lie theory
• Rings and C*-algebras generated by commutators
• Simplicity, bounded normal generation, and automatic continuity of groups of unitaries
• Sums of commutators in pure C*-algebras
• Finite sums of commutators

Cited by

• Commutativity of operator algebras

Related

• Prime ideals in C*-algebras and applications to Lie theory
• Semiprime ideals in C*-algebras
• Lie ideals in properly infinite C*-algebras
• On the Lie ideals of C*-algebras
• C-Ideals of Lie Algebras