On Algebras Generated by Inner Derivations

Abstract

We look for an effective description of the algebra DLie(X , B) of operators on a bimodule X over an algebra B, generated by all operators x → ax − xa, a ∈ B. It is shown that in some important examples DLie(X , B) consists of all elementary operators x → ∑ i aixbi satisfying the conditions ∑ i aibi = ∑ i biai = 0. The Banach algebraic versions of these results are also obtained and applied to the description of closed Lie ideals in some Banach algebras, and to the proof of a density theorem for Lie algebras of operators on Hilbert space.

Cite this paper

@inproceedings{Shulman2008OnAG, title={On Algebras Generated by Inner Derivations}, author={T. V. Shulman and Victor S. Shulman}, year={2008} }