A decomposition theorem for Lie ideals in nest algebras

Abstract

Let N be a nest and let L be a weakly closed Lie ideal of the nest algebra T (N ). We explicitly construct the greatest weakly closed associative ideal J (L) contained in L and show that J (L) ⊆ L ⊆ J (L) ⊕D(L), whereD(L) is an appropriate subalgebra of the diagonal D(N ) of the nest algebra T (N ). We show that norm-preserving linear extensions of elements of the dual of L, satisfying a certain condition, are uniquely determined on the diagonal of the nest algebra by the ideal J (L). Mathematics Subject Classification (2010). Primary 47L35; Secondary 46K50, 17B60.

Weakly closed Lie modules of nest algebras

“…When L is a weakly closed Lie ideal, K L (L), K D (L), K ∆ (L) = {0}. In this situation, it has been shown in [1] that there exists a certain unital weakly closed * -subalgebra D(L) of D K(L) such that K(L) ⊆ J (L) ⊆ L ⊆ K(L) + D K(L) = J (L) ⊕ D(L).…”

“…Although the map φ in the next lemma be generally presented, its definition is in fact rooted in the investigation of the structure of weakly closed Lie T (N )-modules initiated in [1]. Lemma 2.7.…”