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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.

Keywords:  Banach bimodule, Lie submodule, inner derivation, projective tensor product, Varopoulos algebra

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