ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 46A40 Danet, Nicolae, Technical University of Civil Engineering of Bucharest, Romania: Subspaces in directed vector spaces. A directed vector space is an ordered vector space X such that X = X_{+}−X_{+}. A vector subspace G ⊂ X is called: a) directed vector subspace, if a ∈ G, x ∈ X, and 0, a ≤ x, then there exists b ∈ G such that 0, a ≤ b ≤ x; b) ideal, if G is a directed vector subspace with the positive heredity property (0 ≤ x ≤ a ∈ G, x ∈ X ⇒ x ∈ G); c) D-ideal, if G has the positive heredity property and G = G_{+} − G_{+}. These are the analogous notions for vector sublattice and ideal from a vector lattice. The relations between these types of subspaces are studied. Examples, counterexamples and applications to some spaces of operators are presented. Reference: N. Danet, On the notion of ideal in a directed vector space, Rev. Roumaine Math. Pures Appl., 40 (1995), 735-742.