Advanced Search

Let $A$ and $B$ be two Archimedean vector lattices and let ${\left(A^{\text{'}}\right)}_n^{\text{'}}$ and ${\left(B^{\text{'}}\right)}_n^{\text{'}}$ be their order continuous order biduals. If $\Psi :A\times A\to B$ is a positive orthosymmetric bimorphism, then the triadjoint ${\Psi }^{***}:{\left(A^{\text{'}}\right)}_n^{\text{'}}\times {\left(A^{\text{'}}\right)}_n^{\text{'}}\to {\left(B^{\text{'}}\right)}_n^{\text{'}}$ of $\Psi$ is inevitably orthosymmetric. This leads to a new and short proof of the commutativity of almost $f$-algebras.

In the paper we prove that every orthosymmetric lattice bilinear map on the cartesian product of a vector lattice with itself can be extended to an orthosymmetric lattice bilinear map on the cartesian product of the Dedekind completion with itself. The main tool used in our proof is the technique associated with extension to a vector subspace generated by adjoining one element. As an application, we prove that if $(A,*)$ is a commutative $d$-algebra and $A^{𝔡}$ its Dedekind completion, then, $A^{𝔡}$ can be equipped...

In this paper we prove that the image of a nth order derivation on real commutative Banach ℓ-algebras with positive squares is contained in the set of nilpotent elements.