Structure of quasi ordered ∗-vector spaces

Abstract

Let (𝑋,𝑋+) be a quasi ordered ∗-vector space with order unit, that is, a ∗-vector space 𝑋 with order unite together with a cone 𝑋+⊆𝑋. Our main goal is to find a condition weaker than properness of 𝑋, which suffices for fundamental results of ordered vector space theory to work. We show that having a non-empty state space or equivalently having a non-negative order unit is a suitable replacement for properness of 𝑋+. At first, we examine real vector spaces and then the complex case. Then we apply the results to self adjoint unital subspaces of unital ∗-algebras to find direct and shorter proofs of some of the existing results in the literature.