Noncommutative Positivstellensätze for pairs representation-vector

Abstract

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