Structure of quasi ordered ∗-vector spaces

Let (??,??+) be a quasi ordered ?-vector space with order unit, that is, a ?-vector space ?? with order unite together with a cone ??+???. Our main goal is to find a condition weaker than properness of ??, 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 ??+. 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.