Fejér-riesz Factorization Achieved for Positive Noncommutative Trigonometric Polynomials on Discrete Groups

Positive trigonometric polynomials play a crucial role in many areas of mathematics, and Igor Klep from Ecole Polytechnique, Jacob Levenson, and Scott McCullough demonstrate a significant advance in understanding these functions within a noncommutative framework. The researchers prove a Fejér-Riesz type factorization for these polynomials, extending the classical result to more complex settings involving free semigroups and free product groups. This achievement establishes a fundamental connection between the structure of these polynomials and their positivity properties, with implications for verifying inequalities and developing new tools in areas like operator algebras and free probability. Furthermore, the team’s work yields a particularly strong result for a simple case, establishing a powerful Positivstellensatz without the usual requirement of strict positivity, and opens avenues for further investigation into matrix completion problems within these algebraic structures.

Representations of W × Y allow for the construction of A = B*B, where B is analytic and possesses a W-degree of at most w; this degree bound is optimal, and strict positivity is essential for its validity. As a direct application of these findings, the research establishes degree-bounded sums-of-squares certificates for Bell-type inequalities within the context of C[Z *g 2 × Z *h 2] from quantum information theory. In the specific instance where Y equals Zh, the study recovers, within a matrix-valued framework, the classical commutative multivariable Fejér, Riesz factorization.

Positive Definite Functions and Matrix Completion

This document introduces a research project exploring positive definite functions, matrix completions, and related concepts in functional analysis and operator theory. It outlines the core findings, methodology, and broader context of the research, intended for mathematicians and researchers familiar with these areas. The study focuses on positive definite functions, crucial in harmonic analysis, probability, and control theory, and matrix completions, a problem with applications in statistics, machine learning, and signal processing. Researchers investigate these concepts within the framework of non-commutative analysis, specifically utilizing free analysis.

The research incorporates concepts from unitary representations, sums of squares, and the Parrott theorem, a key result related to extending positive definite functions. The work has connections to control theory, the study of how to control dynamical systems. It details the mathematical notation used throughout the research, presents the main results, and explains the organization of the paper. It explores functions that are almost positive definite, and highlights the importance of the Parrott theorem in group theory and positive definite functions.

Fejér-Riesz Factorization for Noncommutative Polynomials

Scientists have achieved a groundbreaking Fejér-Riesz factorization for positive, matrix-valued noncommutative trigonometric polynomials defined on specific domains, advancing the field of free analysis and noncommutative function theory. This work rigorously proves that such polynomials on a domain constructed from either a free semigroup or a free product group, combined with a discrete group, can be factored in a precise manner. The factorization relies on ordering the elements of the domain using the shortlex order, crucial for the proof’s validity. Researchers demonstrate that if a polynomial has degree at most ‘w’ in the variables, and is uniformly strictly positive across all unitary representations, then it can be expressed as a product of analytic polynomials with degree at most ‘w’; this degree bound is optimal, and the requirement of strict positivity is essential for the factorization to hold.

The team developed two novel tools central to this achievement: a positive-semidefinite Parrott theorem with entries defined by functions on a group, and solutions to positive semidefinite matrix completion problems indexed by words in a specific free product group. These tools allow for the controlled completion of matrices, ensuring the positivity required for the factorization. Experiments confirm that the method successfully factors polynomials, delivering degree-bounded sums-of-squares certificates for Bell-type inequalities in a context relevant to quantum information theory. In a special case, the research recovers the classical commutative multivariable Fejér-Riesz factorization in a matrix-valued setting, demonstrating the broad applicability of the new method. Furthermore, scientists obtain a “perfect” group-algebra Positivstellensatz that does not require strict positivity, a result confirmed to be sharp through counterexamples in specific groups. The work establishes a foundational result with implications for diverse areas, including linear systems theory, quantum physics, and free probability, and provides a powerful framework for analyzing noncommutative polynomials and their properties.

Matrix Factorization and Strict Positivity Conditions

Researchers have established a new factorization theorem for trigonometric polynomials with matrix values, extending existing results to encompass free semigroups and free product groups. This achievement demonstrates that under specific conditions, namely, strict positivity across all unitary representations, such polynomials can be expressed as the product of two analytic polynomials with controlled degree. The degree bound attained in this factorization is optimal, highlighting the precision of the result and the essential role of the strict positivity condition. This work builds upon previous findings by providing a more complete and refined understanding of sums-of-squares certificates for Bell-type inequalities, particularly within these algebraic structures.

The team also derived a “perfect” Positivstellensatz for a specific group algebra, removing the need for strict positivity assumptions in that instance, and rigorously demonstrated the limitations of this result through counterexamples in specific groups. The authors acknowledge that while degree bounds are established for the resulting polynomials relative to certain variables, further refinement is needed to determine precise bounds concerning other variables. They also note that the results rely on considering representations on separable Hilbert spaces, a standard approach given the countable nature of the groups involved. Future work could focus on obtaining more concrete estimates for the degree bounds and exploring the implications of these findings for other areas of mathematics and physics.