$$K_0$$-group of absolute matrix order unit spaces

Abstract

In this paper, we describe the Grothendieck group $$K_0(V)$$ of an absolute matrix order unit space V. For this purpose, we discuss the direct limit of absolute matrix order unit spaces. We show that $$K_0$$ is a functor from category of absolute matrix order unit spaces with morphisms as unital completely $$\vert \cdot \vert$$ -preserving maps to category of abelian groups. We study order structure in $$K_0(V)$$ and prove that under certain condition, $$K_0(V)$$ is an ordered abelian group. We also show that the functor $$K_0$$ is additive on orthogonal unital completely $$\vert \cdot \vert$$ -preserving maps.