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.