Estimating the norm of a matrix in $M_n(\mathbb{C})$.

Question

Let $A$ be a $C^*$-algebra and $\tau: A \to \mathbb{C}$ a bounded functional. Let $u = [u_{i,j}] \in M_n(A)$ be a unitary matrix and consider the matrix $m = [\tau(u_{i,j})] \in M_n(\mathbb{C})$. Can we find a good estimate for $\|m\|?$

For example, it would be convenient if I have the following estimate $$\|m\| \le \|\tau\|.$$

Answer 1

Note that $m = (\text{id}_n \otimes \tau)(u)$. Bounded functionals are automatically completely bounded, with $\|\text{id}_n \otimes \tau\| = \|\tau\|$. So now you've got that $$\|m\| =\|\text{id}_n \otimes \tau(u)\| \leq \|\text{id}_n \otimes \tau\|\|u\| = \|\tau\|. $$