On Lie submodules and tensor algebras

Let [graphic not available: see fulltext] be a bimodule over an algebra B, and let [graphic not available: see fulltext] be the algebra of operators on [graphic not available: see fulltext] generated by all operators x ↦ ax − xa, where a ∈ B. We show that in many (but not all) cases, [graphic not available: see fulltext] consists of all elementary operators x ↦ Σaixbi whose coefficients satisfy the conditions Σiaibi = Σibiai = 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.