# Herz-Schur multiplier bounded if corresponding functional is bounded

I want to prove the following statement:

Let $\Gamma$ be a discrete group and $\phi:\Gamma\rightarrow\mathbb{C}$ a function and $\omega_{\phi}:\mathbb{C}[\Gamma]\rightarrow\mathbb{C}:\sum_{t\in\Gamma}\alpha_t t \mapsto \sum_{t\in\Gamma}\alpha_t\phi(t)$ the corresponding functional. If $\omega_{\phi}$ is bounded on $C^*(\Gamma)$ then $\phi$ is a Herz-Schur multiplier with $\|m_{\phi}\|_{cb}\leq \|\phi\|$.

Here $\|m_{\phi}\|_{cb}$ stands for the completely bounded norm of the Schur multiplier $m_{\phi}$ and $C^*(\Gamma)$ is the universal group algebra.

I'm not sure on how to prove this result. I don't quite understand how boundedness of this functional on the universal group algebra can be translated to properties of the multiplier.

• This question comes from appendix D of $C^*$-algebras and finite-dimensional approximations of Brown and Ozawa. In fact they point the reader to the proof of 2.5.11 (3) $\Rightarrow$ (4) to prove this statement, I still didn't find this. – Mathematician 42 May 28 '15 at 17:40