The Arens Triadjoints of some Bilinear Maps

Ruşen Yılmaz

In this paper we study the Arens triadjoints of some bilinear maps on vector lattices. In particular, we prove that, for Archimedean vector lattices $A$ and $B$, the Arens triadjoint i) $T^{***}:A"\times A"\to B"$ of a positive orthosymmetric bilinear map $T:A\times A\to B$ is positive orthosymmetric, and ii) $T^{***}:A"\times A"\to A"$ of a bi-orthomorphism $T:A\times A\to A$ is a bi-orthomorphism. These generalize results on the order bidual of $f$-algebras and almost $f$-algebras in [4].