Springer Iranian Journal of Science 2731-8095 38 4 2014 12 22 Structure of quasi ordered ∗-vector spaces 445 453 2561 10.22099/ijsts.2014.2561 EN G. H. Esslamzadeh Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran M. Moazami Goodarzi Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran F. Taleghani Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran Journal Article 2013 10 19   <span style="font-size: xx-small;">Let </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">(𝑋,𝑋</span></span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">+</span></span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">) </span></span><span style="font-size: xx-small;">be a quasi ordered </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">∗</span></span><span style="font-size: xx-small;">-vector space with order unit, that is, a </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">∗</span></span><span style="font-size: xx-small;">-vector space </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">𝑋 </span></span><span style="font-size: xx-small;">with order unite together with a cone </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">𝑋</span></span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">+</span></span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">⊆𝑋</span></span><span style="font-size: xx-small;">. Our main goal is to find a condition weaker than properness of </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">𝑋</span></span><span style="font-size: xx-small;">, which su</span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">ffi</span></span><span style="font-size: xx-small;">ces 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 </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">𝑋+</span></span><span style="font-size: xx-small;">. At first, we examine real vector spaces and then the complex case. Then we apply the results to self adjoint unital subspaces of unital </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="KO">∗</span></span><span style="font-size: xx-small;">-algebras to find direct and shorter proofs of some of the existing results in the literature. </span> https://ijsts.shirazu.ac.ir/article_2561_38b0fd0cff8b3aaffb439bd8ee4cc2e6.pdf