On algebras generated by inner derivations
Operator Algebras
2008-07-17 v2 Functional Analysis
Abstract
We look for an effective description of the algebra D_{Lie}(X,B) of operators on a bimodule X over an algebra B, generated by inner derivations. It is shown that in some important examples D_{Lie}(X,B) consists of all elementary operators x\to \sum_i a_ixb_i satisfying the conditions $\sum_i a_ib_i = \sum_i b_ia_i = 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
@article{arxiv.0801.4862,
title = {On algebras generated by inner derivations},
author = {Tatiana Shulman and Victor Shulman},
journal= {arXiv preprint arXiv:0801.4862},
year = {2008}
}