Geodesic
On Lie Submodules and Tensor Algebras
Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 91-96

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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. 
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     author = {V. S. Shulman and T. V. Shulman},
     title = {On {Lie} {Submodules} and {Tensor} {Algebras}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {91--96},
     publisher = {mathdoc},
     volume = {43},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a10/}
}
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V. S. Shulman; T. V. Shulman. On Lie Submodules and Tensor Algebras. Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 91-96. http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a10/