Radial multipliers on reduced free products of operator algebras

TL;DR

This paper extends the theory of radial multipliers to reduced free products of operator algebras, establishing conditions for their existence and uniqueness based on trace-class Hankel matrices.

Contribution

It generalizes Wysoczański's result on Herz-Schur multipliers to a broader setting involving free products of operator algebras.

Findings

Existence of unique completely bounded radial multipliers under trace-class Hankel matrix condition

Generalization of Herz-Schur multiplier results to free product operator algebras

Extension of multiplier theory in non-commutative harmonic analysis

Abstract

Let A_i be a family of unital C*-algebras, respectively, of von Neumann algebras and phi: N_0 \to C. We show that if a Hankel matrix related to phi is trace-class, then there exists a unique completely bounded map M_phi on the reduced free product of the A_i, which acts as an radial multiplier. Hereby we generalize a result of Wysocza\'nski for Herz-Schur multipliers on reduced group C*-algebras for free products of groups.