Weak*–closed Jordan ideals of nest algebras

Abstract

Nest algebras provide examples of partial Jordan *–triples. If A is a nest algebra and As = A ∩ A*, where A* is the set of the adjoints of the operators lying in A, then (A, As) forms a partial Jordan *–triple. Any weak*–closed ideal in the nest algebra A is also an ideal in the partial Jordan *–triple (A, As). An analysis of the ideal structure of (A, As) shows that, for a large class of nest algebras, the converse is also true.

Lie Modules of Banach Space Nest Algebras

“…Accordingly, let g j ∈ M ⊥ j , where M j ∈ N and (M j ) + > Nf . By the above claim, each g j ⊗ y ∈ C. Hence, by (3), for all z ∈ N y , the operator g j ⊗ z lies in L. Since L is weakly closed and (g j ⊗ z) weakly converges to g ⊗ z, we have that g ⊗ z ∈ L. Therefore, f ⊗ y ∈ C, as required.…”

“…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.…”

Lie and Jordan Ideals in Reflexive Algebras

“…The converse problem, which has been studied for more than 50 years, is to describe Jordan ideals in terms of associative ideals, and to find conditions under which a non-zero Jordan ideal is in fact an associative ideal, or at least it contains a non-zero ideal. For a few references, see [2,3,27,12,13,28,29,7] and the literature therein. Among other papers, Oliveira in [27] showed that a weakly closed Jordan ideal in a nest algebra, which satisfies certain conditions, is an associative ideal.…”

“…For a few references, see [2,3,27,12,13,28,29,7] and the literature therein. Among other papers, Oliveira in [27] showed that a weakly closed Jordan ideal in a nest algebra, which satisfies certain conditions, is an associative ideal. In Section 4, we extend Oliveira's result quite significantly.…”