On algebras generated by inner derivations

TL;DR

This paper characterizes the algebra generated by inner derivations on bimodules, providing explicit descriptions in key cases and extending results to Banach algebras, with applications to Lie ideals and operator algebras.

Contribution

It offers a new effective description of the algebra generated by inner derivations, including Banach algebraic versions and applications to Lie ideals and density theorems.

Findings

D_{Lie}(X,B) characterized by elementary operators satisfying specific sum conditions

Banach algebraic versions of the main results obtained

Applications to closed Lie ideals and density theorems in operator algebras

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.