On Tractability and Ideal Problem in Non-Associative Operator Algebras

Abstract

The question of the existence of non-trivial ideals of Lie algebras of compact operators is considered from different points of view. One of the approaches is based on the concept of a tractable Lie algebra, which can be of interest independently of the main theme of the paper. Among other results it is shown that an infinite-dimensional closed Lie or Jordan algebra of compact operators cannot be simple. Several partial answers to Wojtyński's problem on the topological simplicity of Lie algebras of compact quasinilpotent operators are also given.

On Simplicity of Lie Algebras of Compact Operators: A Direct Approach

“…A long-standing open question that has called the attention of many mathematicians is the Wojtyński problem [13, Question 3]: Does every closed Lie algebra of compact quasinilpotent operators on a Banach space contain a nontrivial closed Lie ideal? Several partial results centered around this question have been obtained by Brešar, Shulman, and Turovskii [1]. Historically, to the best of our knowledge, consideration of the adjoint representation of elements of Lie algebras in the study of the simplicity of Lie algebras seemed inevitable.…”