ON ALGEBRAS GENERATED BY INNER DERIVATIONS

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 $\mathrm{x}\to \sum _{\mathrm{i}}{\mathrm{a}}_{\mathrm{i}}\mathrm{x}{\mathrm{b}}_{\mathrm{i}}$ satisfying the conditions $\sum _{\mathrm{i}}{\mathrm{a}}_{\mathrm{i}}{\mathrm{b}}_{\mathrm{i}}=\sum _{\mathrm{i}}{\mathrm{b}}_{\mathrm{i}}{\mathrm{a}}_{\mathrm{i}}=0$. 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.