Multiplier norm vs cb norm

Question

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 $m_f$:

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

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

Is there anything else known about the relation of these two norms? For instance, are there examples of $f$ with $\Vert m_f\Vert_{cb}=1$ and $\Vert m_f \Vert_m\to 0$? 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.

Answer 1

The norms are equal if $G$ is amenable, and inequivalent if $G$ is e.g. the free group. Losert has stated that the two norms can be equivalent (in fact, equal IIRC) when $G=SL(2,R)$ but to my knowledge this has never been published

I can't give an explicit reference off the top of my head, but key words here are "Herz-Schur multiplier" and "multipliers of the Fourier algebra". A good place to find out some of the basic terminology, and to find links to the literature, is this set of minicourse notes by Todorov. See Theorem 3.6 in particular.