Fejér--Riesz factorization for positive noncommutative trigonometric polynomials

We prove a Fejér-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 Fejér-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 $\le w$.

💡 Research Summary

This paper establishes a far-reaching generalization of the classical Fejér-Riesz factorization theorem to the setting of noncommutative (nc) trigonometric polynomials. The classical theorem states that a scalar-valued trigonometric polynomial positive on the unit circle can be factored as the modulus squared of an analytic polynomial. The authors extend this result to matrix-valued polynomials defined over domains that mix noncommuting variables (from free structures) and commuting variables (from discrete groups).

The central objects of study are matrix-valued trigonometric polynomials A defined on a product domain ℓ-Frac(W) × Y. Here, W is either the free semigroup on g generators, <x>_g, or the free product group of g copies of Z_2, denoted Z_2^{*g}. The group Y is a discrete group satisfying a specific condition (ℓ-Frac Y = Y), which includes examples like the free abelian group Z^h and the free product group Z_2^{*h}. The degree of a polynomial in the W variables is measured using the shortlex order on words.

The main result, Theorem 1.1, is an nc Fejér-Riesz theorem. It states that if such a polynomial A has W-degree at most w and is uniformly strictly positive on all unitary representations of the group ℓ-Frac(W) × Y on separable Hilbert spaces, then it admits a factorization A = B*B. Here, B is an analytic polynomial (its support is in W × Y, not ℓ-Frac(W) × Y) whose W-degree is also at most w. This degree bound is shown to be optimal. The authors emphasize that the strict positivity assumption is essential; mere positivity is insufficient to guarantee a factorization with such a degree bound, as evidenced by known counterexamples in quantum information theory.

A notable special case occurs when Y is the trivial group. For W = Z_2^{*g}, Theorem 1.6 provides a “perfect” Positivstellensatz: if A is positive semidefinite on all unitary representations, it still factors as B*B with deg_W(B) ≤ deg_W(A), without requiring strict positivity. The paper demonstrates that this perfect result is sharp and unique to Z_2^{*g}; it fails for other free products like Z_2 * Z_3 and Z_3^{*2}, for which counterexamples are constructed in Section 8.

The proofs of these main theorems rely on two novel technical tools developed in the paper, each of independent interest:

1. A positive semidefinite Parrott theorem (Theorem 3.1), where the block entries of the matrix are functions on a group G. This extends the classical Parrott lemma on norm-preserving completions to a positivity-preserving setting in a group context.
2. Solutions to psd matrix completion problems for the free semigroup <x>_g (Theorem 4.1) and the free product group Z_2^{*g} (Theorem 4.2). These completions are for kernels indexed by words of length at most w and have a specific structure dictated by the group’s multiplication.

The paper bridges these matrix completion results with representation theory via a GNS-type construction (Proposition 5.1). A key technical step (Lemma 6.4) involves obtaining a uniform bound on the dimension needed for certain representations, which ultimately leads to the existence of the factor B with a finite degree bound in the Y variables as well (though this bound is not explicitly quantified).

As a significant application, the authors note that their Theorem 1.1 provides degree-bounded sum-of-squares certificates for Bell-type inequalities in the group algebra `C