Squares of Riesz Spaces

A Note on bilinear maps on vector lattices

“…Subsequent developments have been made as a result of contributions by the same authors [9], G. Buskes and A. G. Kusraev [8], and M. A. Toumi [14]. In [14] it is proved that if A, B are vector lattices, (A ′ ) ′ n , (B ′ ) ′ n are their respective order continuous biduals and T : A × A → B is a positive orthosymmetric bilinear map, then the triadjoint T * * * :…”

Order bounded orthosymmetric bilinear operator

“…It is only recently that the class of such operators have been getting more attention, see [8], [11]. A number of important properties of such operators was revealed.…”

“…The same authors in [10] proved that every positive orthosymmetric bilinear operator defined on a sublattice of an f -algebra can be factored through a positive linear operator and the algebra multiplication. These results gave rise to the concept of the square of a vector lattice, developed in [11]. Recently, G. Buskes and A. G. Kusraev in [8] proved that all orthosymmetric order bounded bilinear operators from E × E to the relatively uniformly complete vector lattice F can be represented as compositions of order bounded linear operators from E ⊙ the square of E to F with the canonical bimorphism.…”

Square means and geometric means in lattice-ordered groups and vector lattices

“…In fact, for certain Riesz spaces (E, +, · , ≥), the modulus allows a new addition ⊞ and a new scalar multiplication ⊡ to be defined on the positive cone, E ≥ , of E in such a way that E ≥ becomes the positive cone of a new Riesz space (E , ⊞, ⊡, ≥) (see [3], [5], and [6]). …”

“…The construction of (E , ⊞, ⊡, ≥) in [3] is not elementary in the sense that it requires an equivalence relation on E ≥ × E ≥ , and, although the construction in [6] does not require the use of equivalence relations, it does require a very strong condition on E, viz., uniform completeness. Of course, constructions based on equivalence relations are not unusual.…”