Radial multipliers on arbitrary amalgamated free products of finite von Neumann algebras

Let $(M_i)_{i}$ be a (finite or infinite) family of finite von Neumann algebras with a common subalgebra $P$. When $\varphi:\IN\rightarrow\IC$ is a function, we define the radial multiplier $M_\varphi$ on the amalgamated free product $M=M_1\free_P M_2\free_P\ldots$ setting $M_{\varphi}(x)=\varphi(n)x$ for every reduced expression $x$ of length $n$. In this paper we give a sufficient condition on $\varphi$ to ensure that the corresponding radial multiplier $M_\varphi$ is a completely bounded map, and moreover we give an upper bound on its completely bounded norm. Our condition on $\varphi$ does not depend on the choice of von Neumann algebras $(M_i)_i$ and $P$. This result extends earlier results by Haagerup and M\"oller, who proved the same statement for free products without amalgamation, and M\"oller showed that the same statement holds when $P$ has finite index in each of the $M_i$.