A Non-commutative Real Nullstellensatz Corresponds to a Non-commutative Real Ideal; Algorithms

TL;DR

This paper extends the classical Real Nullstellensatz to non-commutative *-algebras, providing theoretical insights, algorithms for computing real ideals, and examples where the theorem holds in a non-commutative setting.

Contribution

It develops a non-commutative Real Nullstellensatz, introduces algorithms for real ideal computation, and provides examples demonstrating the theorem's applicability in matrix polynomial *-algebras.

Findings

An algorithm computes the smallest real ideal containing a finite set in finite steps.

Theoretical conditions under which a non-commutative real Nullstellensatz holds.

Examples of non-commutative real ideals in matrix polynomial *-algebras.

Abstract

This article takes up the challenge of extending the classical Real Nullstellensatz of Dubois and Risler to left ideals in a *-algebra A. After introducing the notions of non-commutative zero sets and real ideals, we develop three themes related to our basic question: does an element p of A having zero set containing the intersection of zero sets of elements from a finite set S of A belong to the smallest real ideal containing S? Firstly, we construct some general theory which shows that if a canonical topological closure of certain objects are permitted, then the answer is yes, while at the purely algebraic level it is no. Secondly for every finite subset S of the free *-algebra R<x,x*> of polynomials in g indeterminates and their formal adjoints, we give an implementable algorithm which computes the smallest real ideal containing S and prove that the algorithm succeeds in a finite number of steps. Lastly we provide examples of noncommutative real ideals for which a purely algebraic non-commutative real Nullstellensatz holds. For instance, this includes the real (left) ideals generated by a finite sets S in the *-algebra of n by n matrices whose entries are polynomials in one-variable. Further, explicit sufficient conditions on a left ideal in R<x,x*> are given which cover all the examples of such ideals of which we are aware and significantly more.