Lie subalgebras in a certain operator Lie algebra with involution

Abstract

We show in a certain Lie*-algebra, the connections between the Lie subalgebra G+:= G + G* + [G, G*], generated by a Lie subalgebra G, and the properties of G. This allows us to investigate some useful information about the structure of such two Lie subalgebras. Some results on the relations between the two Lie subalgebras are obtained. As an application, we get the following conclusion: Let A ⊂ B(X) be a space of self-adjoint operators and ℒ:= A ⊕ iA the corresponding complex Lie*-algebra. G+ = G + G* + [G, G*] and G are two LM-decomposable Lie subalgebras of ℒ with the decomposition G+ = R(G+) + S, G = RG + SG, and RG ⊆ R(G+). Then G+ is ideally finite iff RG+:= RG + R*G* + [RG, RG*] is a quasisolvable Lie subalgebra, SG+:= SG + SG* + [SG, SG*] is an ideally finite semisimple Lie subalgebra, and [RG, SG] = [RG*, SG] = {0}.