Radial multipliers on reduced free products of operator algebras

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