Absolute matrix order ideals in absolute matrix order unit spaces

TL;DR

This paper introduces and explores absolute matrix order ideals within absolute matrix order unit spaces, establishing their properties and their relation to Grothendieck groups, advancing the understanding of their algebraic structure.

Contribution

It defines absolute matrix order ideals and demonstrates their properties and connection to Grothendieck groups in absolute matrix order unit spaces.

Findings

Construction of absolute matrix order ideals using the absolute matrix order unit property

Grothendieck group of these ideals forms a subgroup of the larger space's Grothendieck group

Provides foundational results for the algebraic structure of absolute matrix order ideals

Abstract

In this paper, we define and study absolute matrix order ideals in absolute matrix order unit spaces. As an application of absolute matrix order unit property, we construct some kinds of absolute matrix order ideals in absolute matrix order unit spaces. Later, we show that the Grothendieck group of a such kind of absolute matrix order ideal for order projections is a subgroup of Grothendieck group of corresponding absolute matrix order unit space for order projections.