2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/106057An analogue of Krein's extension theorem is proved for operator-valued positive definite functions on free groups. The proof gives also the parametrization of all extensions by means of a generalized type of Szego parameters. One singles out a distinguished completion, called central, which is related to quasi-multiplicative positive definite functions. An application is given to factorization of noncommutative polynomials.Functional AnalysisOperator AlgebrasPrimary 43A35; Secondary 05C50, 43A65, 47A57, 47A20text