Journal of Operator Theory

Volume 65, Issue 2, Spring 2011 pp. 281-305.

On algebras generated by inner derivations

Authors: Tatiana Shulman (1) and Victor Shulman (2)
Author institution: (1) Department of Mathematical Sciences, University of Copenhagen, Copenhagen, 2100, Denmark
(2) Department of Mathematics, Vologda State Technical University, Vologda, 160000, Russia

Summary: We look for an effective description of the algebra

$D_{\mathrm{Lie}}(\X,B)$ of operators on a bimodule

$\X$ over an algebra

$B$ , generated by all operators

$x\to ax-xa$ ,

$a\in B$ . It is shown that in some important examples

$D_{\mathrm{Lie}}(\X,B)$ consists of all elementary operators

$x\to \sum\limits_i a_ixb_i$ satisfying the conditions

$\sum\limits_i a_ib_i = \sum\limits_i b_ia_i = 0$ . 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.

Contents Full-Text PDF