Weakly closed Lie modules of nest algebras

Abstract

Abstract. Let T (N ) be a nest algebra of operators on Hilbert space and let L be a weakly closed Lie T (N ) -module. We construct explicitly the largest possible weakly closedMathematics subject classification (2010): 47L35, 46K50, 17B60.

Kernel maps and operator decomposition

“…Let P ′ , Q be any projections in L such that Φ < Q < P ′ < σ(Ψ ⊥ ). By [18,Lemma 1], P ′⊥ T P ′ ∈ M.…”

“…18,Lemma 1],σ(Ψ ⊥ )T σ(Ψ ⊥ )⊥ ∈ M, and by[19], Lemma 3.3,P ′ z ⊗ P ′⊥ w ∈ T (L), it follows [σ(Ψ ⊥ )T σ(Ψ ⊥ ) ⊥ , P ′ z ⊗ P ′⊥ w] = σ(Ψ ⊥ )T σ(Ψ ⊥ ) ⊥ (P ′ z ⊗ P ′⊥ w) − (P ′ z ⊗ P ′⊥ w)σ(Ψ ⊥ )T σ(Ψ ⊥ ) ⊥ = σ(Ψ ⊥ )T σ(Ψ ⊥ ) ⊥ (P ′ z ⊗ P ′⊥ w)lies in M.…”

Lie Modules of Banach Space Nest Algebras

“…Associative, Jordan, and Lie ideals of nest algebras on Hilbert space have been substantially investigated in the literature (e.g., [1][2][3][4]), whilst in what concerns their module structure, or the corresponding theory in the Banach space setting, the literature is not so abundant. Notwithstanding, some headway in that direction has been made, as in the case of [5][6][7][8][9], for example.…”

“…Almost two decades later, this characterisation of Lie ideals of nest algebras was shown to admit an extension to Lie modules ( [8]). It was proved in [8] that for any weakly closed Lie module L of a nest algebra A, similarly to ( [2], Theorem 12), there exist weakly closed A-bimodules J and K, and a von Neumann subalgebra D K of the diagonal of the nest algebra, such that:…”

Bimodules of Banach space nest algebras

“…Finite rank operators play a decisive role in the theory of Hilbert space nest algebras and have been thoroughly investigated with respect to their density and decomposability (e.g., [12,19,20,22,23]). In fact, they have shown to be crucial in the characterisation of associative, Jordan and Lie modules of nest algebras on Hilbert space (e.g., [17,18,23,24]). Still, more recently, they have been given a prominent place in the related context of triangularizability (e.g., [6]).…”