Positive cones on algebras with involution

TL;DR

This paper introduces positive cones on algebras with involution, enabling new algebraic and topological results, including analogues of classical theorems and the characterization of positive involutions.

Contribution

It defines positive cones on algebras with involution and explores their properties, providing foundational tools for algebraic and topological analysis.

Findings

Established the spectral space structure of positive cones

Proved analogues of Artin's solution to Hilbert's 17th problem

Solved the existence problem of positive involutions

Abstract

We introduce positive cones on algebras with involution. These allow us to prove analogues of Artin's solution to Hilbert's 17th problem, the Artin-Schreier theorem characterizing formally real fields, and to define signatures with respect to positive cones. We consider the space of positive cones of an algebra with involution and investigate its topological properties, showing in particular that it is a spectral space. As an application we solve the problem of the existence of positive involutions.