On algebras generated by inner derivations

2008/01/31 by Shulman, Tatiana, Shulman, Victor
#46H25 #47L70 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.0801.4862

Abstract

We look for an effective description of the algebra D_Lie(X,B) of operators on a bimodule X over an algebra B, generated by inner derivations. It is shown that in some important examples D_Lie(X,B) consists of all elementary operators x→ ∑_i a_ixb_i satisfying the conditions ∑_i a_ib_i = ∑_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.

Related

• The derivation problem for one type of bimodules over group algebras
• On Lie and associative algebras containing inner derivations
• Characterizations of 2-local derivations and local Lie derivations on some algebras
• Additive Lie (ξ-Lie) Derivations and Generalized Lie (ξ-Lie) Derivations on Prime Algebras
• Derivations with values in quasi-normed bimodules of locally measurable operators