Let $f:G\to \mathbb{C}$ be a finitely supported functions and let $m_f$ denote the associated multiplier on $C^*_r(G)$, the reduced group $C^*$-algebra:
$$m_f(\alpha)(g)=f(g)\alpha(g)$$
for every $\alpha\in C^*_r(G)$.

We can look at two natural norms of $f$:

- the completely bounded norm $\Vert f\Vert_{cb}$ of $f$ as a multiplier on $C^*_r(G)$
- the multiplier norm $\Vert f\Vert_m$ of $f$ viewed as a bounded linear operator on $C^*_r(G)$.

In general, 
$$\Vert f\Vert_m\le \Vert f\Vert_{cb}.$$

Is there anything else known about the relation of these two norms? For instance, are there examples with $\Vert f\Vert_{cb}=1$ and $\Vert f \Vert\to 0$? Or should they be equal in some cases? I would be grateful for either an answer or direction where to look for this in the literature.