Algebras of Fractions and Strict Positivstellens\"atze for *-Algebras

TL;DR

This paper develops a framework using *-algebras of fractions to establish noncommutative Positivstellens"atze, providing new results for Weyl and affine group Lie algebras and criteria for their representations.

Contribution

It introduces a novel approach with *-algebras of fractions to prove noncommutative Positivstellens"atze and applies it to Weyl and affine group Lie algebras.

Findings

New strict Positivstellens"atze for Weyl algebra

Characterization of integrable representations of the affine Lie algebra

A new integrability criterion for Lie algebra representations

Abstract

In this paper we investigate a *-algebra $\cX$ of fractions associated with a unital complex *-algebra $\cA$. The algebra $\cX$ and its Hilbert space representations are used to prove abstract noncommutative strict Positivstellens\"atze for $\cA$. Multi-grading of $\cA$ are studied as technical tools to verify the assumptions of this theorem. As applications we obtain new strict Positivstellens\"atze for the Weyl algebra and for the Lie algebra $\cg$ of the affine group of the real line. We characterize integrable representations of the Lie algebra $\cg$ in terms of resolvents of the generators and derive a new integrability criterion for representations of $\cg$.