When All Separately Band Preserving Bilinear Operators Are Symmetric ?

The aim of this note is to give an algebraic characterization of those universally complete vector lattice in which all band preserving bilinear operators are symmetric. We start with recalling some definitions and auxiliary facts about bilinear operators on vector lattices. For the theory of vector lattices and positive operators we refer to the books [1] and [7]. Let E and F be vector lattices. A bilinear operator b : E×E → G is called orthosymmetric if x ∧ y = 0 implies b(x, y) = 0 for arbitrary x, y ∈ E, see [3]. Recall also that b is said to be symmetric (or antisymmetric) if b(x, y) = b(y, x) (respectively b(x, y) = −b(y, x)) for all x, y ∈ E. Finally, b is said to be positive if b(x, y) > 0 for all 0 6 x, y ∈ E and orthoregular if it can be represented as the difference of two positive orthosymmetric bilinear operators [2]. The vector space of all orthoregular bilinear operators and its subspaces are always considered with the ordering determined by the cone of positive operators. The following important property of orthosymmetric bilinear operators is due to G. Buskes and A. van Rooij (see [3, Corollary 2]):