Radial multipliers on reduced free products of operator algebras

Let AiAi be a family of unital C¿C¿-algebras, respectively, of von Neumann algebras and ¿:N0¿C¿:N0¿C. We show that if a Hankel matrix related to ¿ is trace-class, then there exists a unique completely bounded map M¿M¿ on the reduced free product of the AiAi, which acts as a radial multiplier. Hereby we generalize a result of Wysoczanski for Herz–Schur multipliers on reduced group C¿C¿-algebras for free products of groups.