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Noncommutative Positivstellensätze for pairs representation-vector

We study non-commutative real algebraic geometry for a unital associative *-algebra A viewing the points as pairs (π,v) where π 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. Schmüdgen's) non-commutative real algebraic geometry where the points are unbounded *-representation of A.