The orthomodular identity and metric completeness of the coordinatizing division ring

Abstract:

Let $F$ be any division subring of the real quaternions $H$. Let ${l_2}(F)$ denote the linear space of all square summable sequences from $F$ and let $L$ denote the lattice of all “$\bot$-closed” subspaces of ${l_2}(F)$, where “$\bot$” denotes the orthogonality relation derived from the $H$-valued form $(a,b) = \sum \nolimits _{i = 1}^\infty {{a_i}{{\overline b }_i}}$ where $a,b \in {l_2}(F),a = ({a_i};i = 1,2, \cdots )$ and $b = ({b_i};i = 1,2, \cdots )$. Then $L$ is complete, orthocomplemented, $M$-symmetric, irreducible, atomistic, and separable, but $L$ is orthomodular if and only if $F$ is either the reals, the complex numbers, or the quaternions.