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Fully noncentral Lie ideals and invariant additive subgroups in rings

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arxiv 2409.03362 v2 pith:SAISBNEK submitted 2024-09-05 math.RA math.OA

Eusebio Gardella , Tsiu-Kwen Lee , Hannes Thiel This is my paper · ORCID
classification math.RAmath.OA
keywords additivenoncentralfullyinvarianteveryidealsubgroupalgebra
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abstract

We prove conditions ensuring that a Lie ideal or an invariant additive subgroup in a ring contains all additive commutators. A crucial assumption is that the subgroup is fully noncentral, that is, its image in every quotient is noncentral. For a unital algebra over a field of characteristic $\neq 2$ where every additive commutator is a sum of square-zero elements, we show that a fully noncentral subspace is a Lie ideal if and only if it is invariant under all inner automorphisms. This applies in particular to zero-product balanced algebras.

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  1. Lie ideals in properly infinite C*-algebras

    math.OA 2024-12 accept novelty 7.0 of 10

    Every Lie ideal in a unital, properly infinite C*-algebra is commutator equivalent to a unique two-sided ideal, and the same uniqueness holds in von Neumann algebras without a commutative summand.

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