Noncommutative Positivstellens\"atze for pairs representation-vector

TL;DR

This paper develops noncommutative Positivstellensatz results for pairs of representations and vectors in *-algebras, extending classical noncommutative real algebraic geometry to new settings involving pairs.

Contribution

It introduces a framework for noncommutative real algebraic geometry focused on pairs of *-representations and vectors, broadening the scope of Positivstellensatz results.

Findings

Established Positivstellensatz for pairs of representations and vectors.

Compared new results with classical noncommutative algebraic geometry.

Extended the theory to matrix polynomial *-algebras.

Abstract

We study non-commutative real algebraic geometry for a unital associative *-algebra A viewing the points as pairs ({\pi},v) where {\pi} is an unbounded *-representation of A on an inner product space which contains the vector v. We first consider the *-algebras of matrices of usual and free multivariate polynomials with their natural subsets of points. If all points are allowed then we can obtain results for general A. Finally, we compare our results with their analogues in the usual (i.e. Schm\"udgen's) non-commutative real algebraic geometry where the points are unbounded *-representation of A.