Fejer-Riesz factorization for free products
Abstract
We establish an operator-valued Fejer-Riesz factorization theorem for the algebraic free product A of C[F], the group algebra on the free group F on finitely many generators, with finitely many finite dimensional C-star algebras. With respect to a natural reduced-word length, called extent, we show that every linear combination F of elements of A with coefficients from the algebra of operators on a Hilbert space of extent at most 2d that is positive under all finite-dimensional unital *-representations of A admits a factorization F=B^*B with B of extent at most d. This sharp half-extent bound holds without strict positivity. The proof combines a new finite-dimensional realization theorem for positive truncated kernels with operator-system methods. We also establish factorization results for tensor products with group algebras of arbitrary discrete groups under strict positivity in the maximal C-star-tensor product; for amenable groups, strict positivity in the minimal tensor product suffices.
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BibTeX
@misc{jindal2026fejerriesz,
title = {{Fejer-Riesz factorization for free products}},
author = {Abhay Jindal and Igor Klep and Jacob Levenson and Scott McCullough},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08554},
eprint = {2611.08554},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08554},
}
RIS
TY - UNPB
TI - Fejer-Riesz factorization for free products
AU - Abhay Jindal
AU - Igor Klep
AU - Jacob Levenson
AU - Scott McCullough
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08554
UR - https://arxiv.org/abs/2611.08554
AB - We establish an operator-valued Fejer-Riesz factorization theorem for the algebraic free product A of C[F], the group algebra on the free group F on finitely many generators, with finitely many finite dimensional C-star algebras. With respect to a natural reduced-word length, called extent, we show that every linear combination F of elements of A with coefficients from the algebra of operators on a Hilbert space of extent at most 2d that is positive under all finite-dimensional unital *-representations of A admits a factorization F=B^*B with B of extent at most d. This sharp half-extent bound holds without strict positivity. The proof combines a new finite-dimensional realization theorem for positive truncated kernels with operator-system methods. We also establish factorization results for tensor products with group algebras of arbitrary discrete groups under strict positivity in the maximal C-star-tensor product; for amenable groups, strict positivity in the minimal tensor product suffices.
ER -