$K_1$ and $K$-groups of absolute matrix order unit spaces

TL;DR

This paper develops the theory of Grothendieck groups $K_1(V)$ and $K(V)$ for absolute matrix order unit spaces, establishing their properties, functoriality, and relations to other $K$-groups.

Contribution

It introduces and analyzes the $K_1$ and $K$ groups for absolute matrix order unit spaces, including their ordered structure and functorial properties.

Findings

$K_1(V)$ and $K(V)$ are ordered abelian groups.

They are functors from the category of absolute matrix order unit spaces.

Under certain conditions, $K(V)$ quotient is isomorphic to $K_0(V) igoplus K_1(V).

Abstract

In this paper, we describe the Grothendieck groups $K_1(V)$ and $K(V)$ of an absolute matrix order unit space $V$ 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. The construction of $K(V)$ follows in a almost similar manner as that of $K_1(V).$ We prove that $K_1(V)$ and $K(V)$ are ordered abelian groups. We also prove that $K_1(V)$ and $K(V)$ are functors from the category of absolute matrix order unit spaces with morphisms as unital completely $\vert \cdot \vert$-preserving maps to the category of ordered abelian groups. Later, we show that under certain conditions, quotient of $K(V)$ is isomorphic to the direct sum of $K_0(V)$ and $K_1(V),$ where $K_0(V)$ is the Grothendieck group for order projections.