We show that for a linear space of operators M ⊆ B(H 1 , H 2 ) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ 1 , ψ 2 ) on a bilattice Bil(M) of subspaces determined by M, with P ≤ ψ 1 (P, Q) and Q ≤ ψ 2 (P, Q), for any pair (P, Q) ∈ Bil(M), and such that an operator T ∈ B(H 1 , H 2 ) lies in M if and only if ψ 2 (P, Q)T ψ 1 (P, Q) = 0 for all (P, Q) ∈ Bil(M). This extends to reflexive spaces the Erdos-Power type characterization of weakly closed bimodules over a nest algebra. 2010 Mathematics Subject Classification. Primary 47A15.

Let B be a JBW * -triple, let A be a JB * -subtriple of B and let R(A) be the set of range tripotents relative to A. It is shown that, under certain conditions, the supremum of a family of range tripotents in R(A) coincides with that in the complete latticeŨ(B) of all tripotents in B. As a consequence, a sufficient condition for a tripotent to be a range tripotent relative to A is obtained. The action of isomorphisms on range tripotents is investigated, and an analysis of the suprema of families of spectral range tripotents leads to a generalization of a result known for open projections in W * -algebras. Mathematics Subject Classification (2000). Primary 46L70; Secondary 17C65.

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.

We introduce the notions of kernel map and kernel set of a bounded linear operator on a Hilbert space relative to a subspace lattice. The characterization of the kernel maps and kernel sets of finite rank operators leads to showing that every norm closed Lie module of a continuous nest algebra is decomposable. The continuity of the nest cannot be lifted, in general.2010 Mathematics Subject Classification. 47A15, 47L35, 17B60.

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.