On Lie Submodules and Tensor Algebras

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.