A representation theorem for archimedean quadratic modules on *-rings

TL;DR

This paper generalizes Jacobi's representation theorem for archimedean quadratic modules from commutative rings to noncommutative *-rings using $C^*$-algebra representation theory, extending noncommutative real algebraic geometry.

Contribution

It introduces a noncommutative version of Jacobi's theorem for *-rings, based on the Gelfand-Naimark representation theorem and Fujimoto's noncommutative theory.

Findings

Generalizes Jacobi's theorem to *-rings

Connects noncommutative algebra with $C^*$-algebra representations

Provides a new framework for noncommutative real algebraic geometry

Abstract

We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings, \cite[Theorem 5]{jacobi}. We show that this theorem is a consequence of the Gelfand-Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand-Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.