Positive cones and gauges on algebras with involution

TL;DR

This paper extends classical valuation and ordering theory to finite-dimensional simple algebras with involution, establishing new links and lifting results for gauges and positive cones.

Contribution

It introduces a framework connecting gauges and positive cones on algebras with involution, generalizing classical valuation-ordering relations to a broader algebraic context.

Findings

Established a correspondence between gauges and positive cones on algebras with involution.

Proved lifting theorems analogous to the Baer-Krull theorem for fields.

Extended classical valuation-ordering links to noncommutative algebraic structures.

Abstract

We extend the classical links between valuations and orderings on fields to Tignol-Wadsworth gauges and positive cones on finite-dimensional simple algebras with involution. We also study the compatibility of gauges and positive cones, and prove lifting results in the style of the Baer-Krull theorem for fields.