Prime and semiprime Lie ideals in C*-algebras
Eusebio Gardella, Kan Kitamura, Hannes Thiel
TL;DR
This paper establishes a tight structural bridge between Lie ideals and two-sided ideals in $C^*$-algebras via the full normalizer $T(I)$. Leveraging Dixmier ideals and their powers, it proves that every semiprime Lie ideal arises as $T(I)$ for a semiprime two-sided ideal $I$, and conversely identifies a largest contained ideal $I(L)$ to realize any given semiprime Lie ideal; under suitable conditions, this yields bijections between semiprime/prime two-sided ideals and their corresponding Lie ideals. The authors show that, when the algebra is generated by commutators, the correspondence becomes a complete bijection between semiprime (resp. prime) two-sided ideals and semiprime (resp. prime) Lie ideals, with a unique realizing ideal for each Lie ideal. They also analyze the role of the commutator ideal (notably in von Neumann algebras) and provide a criterion for fully noncentral Lie ideals, culminating in concrete descriptions for unital simple nonabelian algebras where the only semiprime/prime Lie ideals are the center and the whole algebra.
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.