Gauge-Invariant Uniqueness and Reductions of Ordered Groups

TL;DR

This paper establishes a gauge-invariant uniqueness theorem for $P$-graph algebras derived from weakly quasi-lattice ordered groups that reduce to amenable ordered groups, expanding understanding of their structural properties.

Contribution

It introduces conditions under which gauge-invariant uniqueness holds for $P$-graph algebras and characterizes the class of ordered groups reducible to amenable groups.

Findings

Gauge-invariant uniqueness theorem for certain $P$-graph algebras

Class of ordered groups reducing to amenable groups includes all amenable groups

Closure properties of this class under direct products, free products, and hereditary subgroups

Abstract

A reduction $\varphi$ of an ordered group $(G,P)$ to another ordered group is an order homomorphism which maps each interval $[1,p]$ bijectively onto $[1, \varphi(p)]$. We show that if $(G,P)$ is weakly quasi-lattice ordered and reduces to an amenable ordered group, then there is a gauge-invariant uniqueness theorem for $P$-graph algebras. We also consider the class of ordered groups which reduce to an amenable ordered group, and show this class contains all amenable ordered groups and is closed under direct products, free products, and hereditary subgroups.