Brief communications

On Lie Submodules and Tensor Algebras

V. S. Shulman^a, T. V. Shulman<sup>b

a</sup> Vologda State Technical University
^b Department of Mathematical Sciences, University of Copenhagen

Abstract: Let $\mathcal{X}$ be a bimodule over an algebra $B$, and let $\mathcal{D}_{\text{Lie}}(\mathcal{X},B)$ be the algebra of operators on $\mathcal{X}$ generated by all operators $x\mapsto ax-xa$, where $a\in B$. We show that in many (but not all) cases, $\mathcal{D}_{\text{Lie}}(\mathcal{X},B)$ consists of all elementary operators $x\mapsto\sum a_ixb_i$ whose coefficients satisfy the conditions $\sum_i a_ib_i=\sum_ib_ia_i=0$. Analogs of these results are proved for Banach bimodules over Banach algebras. Using them, we obtain the description of the structure of closed Lie ideals for a class of Banach algebras and prove some density theorems for Lie algebras of operators on Hilbert spaces.

Keywords: Banach algebra, derivation, Lie ideal, support of an operator.

UDC: 512.553+517.986.2

Received: 30.07.2007

DOI: 10.4213/faa2941

 English version: Functional Analysis and Its Applications, 2009, 43:2, 158–161

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