On the modulus of disjointness preserving operators on complex vector lattices

Abstract.

Let L and M be Archimedean vector lattices such that $L_\mathbb{C} = L + iL$ and $M_\mathbb{C} = M + iM$ are complex vector lattices. We constructively and intrinsically prove that if $\mathcal{T} = U + iV$ is an order bounded disjointness preserving operator from $L_\mathbb{C}$ into $M_\mathbb{C}$ then the modulus

$$\left| \mathcal{T} \right| = \sup \left\{ {\left( {\cos \theta } \right)U + \left( {\sin \theta } \right)V:\theta \in \left[ {0, 2\pi } \right]} \right\}$$

of $\mathcal{T}$ exists in the ordered vector space of all order bounded operators from L into M.