Characterization of the $\sigma$-Dedekind complete Riesz space by the subadditivity of its positive part mapping

Abstract

Two retractions $M$ and $N$ on convex cones $\mathbf{M}$ and respectively $\mathbf{N}$ of a real vector space $X$ are called mutually polar if $M+N=I$ and $MN=NM=0.$ In this note it is shown, that if the cones $\mathbf{M}$ and $\mathbf{N}$ are generating, $\sigma$-monotone complete, $M$ and $N$ are $\sigma$-monotone continuous, then the subadditivity of $M$ and $N$ (with respect to the order relation induced by $\mathbf{M}$ and respectively $\mathbf{N}$) imply that $\mathbf{M}$ and $\mathbf{N}$ are lattice cones. $(X,\mathbf{M})$ and $(X,\mathbf{N})$ become $\sigma$-Dedekind complete Riesz spaces, $M$ and $-N$ are the positive part , respectively the negative part mappings in $(X,\mathbf{M})$; $N$ and $-M$ are the positive part, respectively the negative part mappings in $(X,\mathbf{N})$.