$K_0$-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 on $K_0(V)$ 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.