Fej\'er--Riesz factorization for positive noncommutative trigonometric polynomials

We prove a Fej\'er-Riesz type factorization for positive matrix-valued noncommutative trigonometric polynomials on $\mathscr{W}\times\mathfrak{Y}$, where $\mathscr{W}$ is either the free semigroup $\langle x \rangle_g$ or the free product group $\mathbb{Z}_2^{g}$, and $\mathfrak{Y}$ is a discrete group. More precisely, using the shortlex order, if $A$ has degree at most $w$ in the $\mathscr{W}$ variables and is uniformly strictly positive on all unitary representations of $\mathscr{W}\times\mathfrak{Y}$, then $A=B^{*}B$ with $B$ analytic and of $\mathscr{W}$-degree at most $w$; this degree bound is optimal, and strict positivity is essential. As an application, we obtain degree-bounded sums-of-squares certificates for Bell-type inequalities in $\mathbb{C}[\mathbb{Z}_2^{*g}\times \mathbb{Z}_2^{*h}]$ from quantum information theory. In the special case $\mathscr{W}=\mathbb{Z}^h$ we recover, in the matrix-valued setting, the classical commutative multivariable Fej\'er-Riesz factorization. For trivial $\mathfrak{Y}$ we obtain a ``perfect'' group-algebra Positivstellensatz on $\mathbb{Z}_2^{*g}$ that does not require strict positivity; this result is sharp, as demonstrated by counterexamples in $\mathbb{Z}_2*\mathbb{Z}_3$ and $\mathbb{Z}_3^{*2}$. To establish our main results two novel ingredients of independent interest are developed: (a) a positive-semidefinite Parrott theorem with entries given by functions on a group; and (b) solutions to positive semidefinite matrix completion problems for $\langle x \rangle_g$ or the free product group $\mathbb{Z}_2^{*g}$ indexed by words in $\mathscr{W}$ of length $\le w$.