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

TL;DR

This paper explores the simplicity of Lie algebras of compact operators on infinite-dimensional Hilbert spaces, providing new insights by linking algebraic properties to operator ideals.

Contribution

It introduces a novel approach to analyzing Lie algebra simplicity using the concept of soft-edged operator ideals, extending previous work on the Wojtyński problem.

Findings

Lie algebras of compact operators are generally not simple under mild conditions

The study connects algebraic simplicity to properties of operator ideals

Provides a new perspective on the structure of Lie algebras in functional analysis

Abstract

We investigate an algebraic variant of the Wojty\'nski problem on the simplicity of Lie algebras of compact operators on a separable infinite-dimensional complex Hilbert space. We prove the non-simplicity of Lie algebras of compact operators under a mild softness condition using the notion of soft-edged operator ideals introduced by Kaftal and Weiss. We believe our study offers a new perspective on the investigation of the simplicity of Lie algebras by relating it to the study of operator ideals.