Real Nullstellensatze and *-ideals in *-algebras

TL;DR

This paper investigates *-ideals in free *-algebras over real or complex fields, establishing Nullstellensatz results for certain classes of ideals, especially those generated by analytic polynomials, and connecting these to broader algebraic theory.

Contribution

It proves Nullstellensatz results for *-ideals generated by analytic polynomials and explores their properties in free *-algebras, extending existing algebraic frameworks.

Findings

Finite codimension real *-ideals satisfy Nullstellensatz.

Analytic polynomial generated *-ideals satisfy Nullstellensatz.

Homogeneous analytic polynomial *-ideals have simple descriptions.

Abstract

Let F denote either the real or complex field. An ideal I in the free *-algebra F<x,x*> in g freely noncommuting variables and their formal adjoints is a *-ideal if I = I*. When a real *-ideal has finite codimension, it satisfies a strong Nullstellensatz. Without the finite codimension assumption, there are examples of such ideals which do not satisfy, very liberally interpreted, any Nullstellensatz. A polynomial p in F<x,x*> is analytic if it is a polynomial in the variables {x} only; that is if p in F<x>. As shown in this article, *-ideals generated by analytic polynomials do satisfy a natural Nullstellensatz and those generated by homogeneous analytic polynomials have a particularly simple description. The article also connects the results here for *-ideals to the literature on Nullstellensatz for left ideals in *-algebras generally and in F<x,x*> in particular. It also develops the concomitant general theory of *-ideals in general *-algebras.