K1 and K -groups of absolute matrix order unit spaces

Abstract

In this paper, we describe the Grothendieck groups K1(X) and K(X) of an absolute matrix order unit space X for unitary and partial unitary elements, respectively. For this purpose, we study some basic properties of unitary and partial unitary elements, and define their path homotopy equivalence. We prove that K1(X) and K(X) are ordered abelian groups. We also prove that K1(X) and K(X) are functors from the category of absolute matrix order unit spaces with morphisms as unital completely absolute value preserving maps to the category of ordered abelian groups. Later, we show that under certain conditions, quotient of K(X) is isomorphic to the direct sum of K(X) and K1(X) , where K(X) is the Grothendieck group for order projections. � 2023, Tusi Mathematical Research Group (TMRG).