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 Wysocza´nski for Herz–Schur multipliers on reduced group C ∗-algebras for free products of groups. © 2012 Elsevier Inc. All rights reserved