Maximal quadratic modules on *-rings

TL;DR

This paper extends the concept of maximal quadratic modules from commutative rings to *-rings, exploring their structure, properties, and applications in noncommutative real algebraic geometry, including an extension of Schmüdgen's Positivstellensatz.

Contribution

It introduces a generalization of maximal quadratic modules to *-rings and establishes their structural properties and applications in noncommutative algebra.

Findings

Support of maximal quadratic modules is a symmetric prime *-ideal.

Maximal quadratic modules in Noetherian *-rings relate to those in simple artinian rings.

Extension of Schmüdgen's Positivstellensatz to the Weyl algebra.

Abstract

We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to $\ast$-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime $\ast$-ideal, that every maximal proper quadratic module in a Noetherian $\ast$-ring comes from a maximal proper quadratic module in a simple artinian ring with involution and that maximal proper quadratic modules satisfy an intersection theorem. As an application we obtain the following extension of Schm\" udgen's Strict Positivstellensatz for the Weyl algebra: Let $c$ be an element of the Weyl algebra $\mathcal{W}(d)$ which is not negative semidefinite in the Schr\" odinger representation. It is shown that under some conditions there exists an integer $k$ and elements $r_1,...,r_k \in \mathcal{W}(d)$ such that $\sum_{j=1}^k r_j c r_j^\ast$ is a finite sum of hermitian squares. This result is not a proper generalization however because we don't have the bound $k \le d$.