On orthosymmetric bilinear maps

Abstract

We use a theorem by Buskes-Kusraev on the automatic symmetry of an order bounded orthosymmetric bilinear map to give a complete description of order bounded derivations on a pseudo f -algebra. This generalizes a well known theorem by Colville, Davis, and Keimel. Then, we investigate the commutator [, ] of an orthosymmetric bilinear map using an order theoretical approach. This leads to a generalization of the Buskes-Kusraev result to a larger class of orthosymmetric bilinear maps.

Characterization of hyper-archimedean vector lattices via disjointness preserving bilinear maps

“…, e n are some components of e. for all a, b ∈ A. By ( [2], Theorem 14), Ψ is symmetric, and Ψ(a, b) = Ψ(a * b, e) for all a, b ∈ A, as required.…”

“…Then Ψ is (r.u) continuous. It follows that, by ( [2], Theorem 14), Ψ is symmetric. Let e ∈ A + and let 0 ≤ |x|, |y| ≤ e in A.…”

“…Since, in this paper, all vector lattices and -algebras under consideration are archimedean, then any f -algebra, in this section, is commutative; see [2,3,6].…”

Product factorability of integral bilinear operators on Banach function spaces

“…The same authors gave the connection between the orthosymmetric bilinear operators and squares and powers of vector lattices (see [11]). Moreover, the commutators of orthosymmetric bilinear maps were recently investigated by Ben Amor, who gave a more general class of orthosymmetric bilinear maps related to the symmetry condition by defining the concept of relatively uniform continuity [6]. Finally, let us report that Erdogan et al have given a factorization theorem for zero product preserving maps defined on the Cartesian product of Hilbert spaces through the convolution product [17].…”

Product factorable multilinear operators defined on sequence spaces

“…For multilinear operators acting in Banach algebras, this factorization gives useful results for the weighted homomorphisms and derivations, where algebraic multiplication is considered as the speci…c map (see [1,2] and references therein). For Riesz spaces, such a factorization is used to obtain interesting results regarding powers of vector lattices, in which orthogonality is involved (see [4,6,7] and references therein).…”