  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.
