Radial multipliers on amalgamated free products of II_1-factors

TL;DR

This paper extends the theory of radial multipliers to amalgamated free products of II_1-factors, establishing conditions for the existence of unique completely bounded maps based on trace-class Hankel matrices.

Contribution

It introduces a new class of radial multipliers on amalgamated free products of II_1-factors, generalizing previous results to a broader algebraic setting.

Findings

Existence of unique completely bounded maps under trace-class Hankel matrix condition.

Extension of Haagerup's results to amalgamated free products.

Conditions linking Hankel matrix properties to multiplier boundedness.

Abstract

Let $\mathcal{M}_i$ be a family of $\mathrm{II}_1$-factors, containing a common $\mathrm{II}_1$-subfactor $\mathcal{N}$, such that $[\mathcal{M}_i:\mathcal{N}] \in \mathbb{N}_0$ for all $i$. Furthermore, let $\phi \colon \mathbb{N}_0 \to \mathbb{C}$. 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 amalgamated free product of the $\mathcal{M}_i$ with amalgamation over $\mathcal{N}$, which acts as an radial multiplier. Hereby we extend a result of U. Haagerup and the author for radial multipliers on reduced free products of unital $C^*$- and von Neumann algebras.