On Some Properties of Bilinear Maps of Order Bounded Variation

Abstract

In this paper we study properties of bilinear maps of order bounded variation. Theorems of preservation of properties in passage to the triadjoint and the tensor product are presented.

On order bounded disjointness preserving operators

“…Some version of Meyer's Theorem is true for bilinear operators: If X, Y , and Z are vector lattices and B : X × Y → Z is an order bounded disjointness preserving bilinear operator then B possesses the positive part B + , the negative part B − , and the modulus |B|, which are lattice bimorphisms; moreover, B + (x, y) = B(x, y) + and B − (x, y) = B(x, y) − for all x ∈ X + and y ∈ Y + , and |B|(|x|, |y|) = |B(x, y)| for all x ∈ X and y ∈ Y ; see [11,Theorem 5] and [21,Theorem 3.4]. In particular, B is regular.…”

An almost orthosymmetric bilinear map

“…The Arens multiplication introduced in [3] on the bidual of various lattice ordered algebras has been well documented (see, e.g., [4]). The more general question about Arens triadjoints of bilinear maps on products of vector lattices has recently aroused considerable interest (see, e.g., [7]). In this direction, as the extensions of the notions of classes of almost f -algebra, f -algebra, d-algebra and pseudo-almost f -algebra, we have studied the Arens triadjoints of some classes of bilinear maps on vector lattices; mainly, orthosymmetric bilinear maps, bi-orthomorphisms, d-bimorphisms and almost orthomorphism bilinear maps (see [15,14]): De…nition 1.…”

A Note on bilinear maps on vector lattices

“…The more general question about Arens triadjoints of bilinear maps on products of vector lattices has recently aroused considerable interest (see, e.g., [7]). In Theorem 2.1 in [7] several properties of the Arens triadjoint maps are collected. For example, the adjoint of a bilinear map of order bounded variation is of order bounded variation and the triadjoint of such a map is separately order continuous.…”