The triadjoint of an orthosymmetric bimorphism

B-Bimorphisms

“…f -algebras, almost f -algebras, d-algebras and b-algebras in lattice ordered algebras were studied and their order biduals and order continuous biduals were investigated by the mathematicians in [3,6,9] using Arens products. The Arens triadjoint of bilinear mappings on products of vector lattices has been investigated by some mathematicians, for example, A. Toumi [8], R. Yilmaz O. Gok [10,11]. In this paper, we study the Arens triadjoints of bilinear mappings which are b-bimorphisms on vector lattices.…”

“…The triadjoint of orthosymmetric bimorphism is also an orthosymmetric bimorphism by Toumi [8]. Toumi's proof is focused that the triadjoint of orthosymmetric bilinear mapping is defined on the order continuous bidual of vector lattice.…”

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 * * * :…”

The Arens triadjoints of some bilinear maps

“…The class of orthosymmetric bilinear maps was introduced in [10] by G. Buskes and A. van Rooij. 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 and B are Archimedean 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 * * * : (A ) n × (A ) n → (B ) n of T is a positive orthosymmetric bilinear map.…”

“…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 and B are Archimedean 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 * * * : (A ) n × (A ) n → (B ) n of T is a positive orthosymmetric bilinear map. In particular, in Section 2 we extend this result to the whole A × A ; that is, if A and B are Archimedean vector lattices, A and B are the order biduals of A and B respectively, then T * * * : A × A → B is a positive orthosymmetric bilinear map whenever T : A × A → B is so.…”