Operator-Valued Positivstellens\"atze on Matrix Convex Sets and Free Products of Finite Abelian Groups

Abstract

We prove a Positivstellensatz for operator-valued noncommutative polynomials that are positive on matrix convex sets. Specifically, let $p$ be an operator-valued polynomial in $B(H)\otimes C $ of degree at most $2d+1$, where $H$ is separable and infinite-dimensional. Let $L(x)=I+\sum_{j=1}^{g} A_j x_j$ be a monic linear operator pencil, and let $D_L=\{X: L(X) \geq 0\}$ be the associated matrix convex set. We show that $p$ is positive on $D_L$ if and only if $p=r^*r+q^*\pi(L)q$, where $q$ and $r$ have degree at most $d$, and $\pi$ is a unital completely positive map on the operator system generated by the coefficients of $L$. The proof combines a Hahn--Banach separation argument with a tailored GNS construction. The main challenge is that the separation occurs in the product ultraweak topology, so boundedness of the resulting GNS operators is not automatic. We first handle bounded matrix convex sets, using closedness of the cone of weighted squares in the product ultraweak topology as the key technical input, and then pass to the general unbounded case by an approximation argument. Finally, we apply this convex Positivstellensatz to prove an operator-valued noncommutative Fejer--Riesz theorem on free products of finite abelian groups. The key additional ingredients are the universal $*$-algebra povm(n) associated with POVMs, a perfect Positivstellensatz for povm(n), and Boca's theorem on free products of completely positive maps. As a consequence, every positive operator-valued trigonometric polynomial on a free product of finite abelian groups admits a sum-of-squares factorization with explicit complexity bounds.