Let $A = M_n(\mathbb C)$. Then it is possible to endow this $\ast$-algebra with several different norms (see here):
$$ \|a\|_1 = \max_j \sum_i |a_{ij}|$$
$$ \|a\|_\infty = \max_i \sum_j |a_{ij}|$$
and the operator norm:
$$ \|a\| = \sup_{\|v\|=1}\|Av\|$$
It is pointed out in the Wikipedia article that the operator norm is different from the $1$- and the $\infty$-norm.
Assuming that all three norms above make $A$ a Banach algebra, I was wondering which of them makes it also a $C^\ast$-algebra? (there can be only one)
On a related note, would $\displaystyle \|a\|_p = \left ( \max_j \sum_i |a_{ij}|^p \right)^{1 \over p}$ and $\displaystyle \|a\|_p = \left ( \max_i \sum_j |a_{ij}|^p \right)^{1 \over p}$ also define matrix norms?
It is the operator norm. It is the only one that satisfies $\|a^*a\|=\|a\|^2$ for all $a$. Actually, you obtain the operator norm precisely when you represent $M_n(\mathbb C)$ as the algebra of linear operators on $\mathbb C^n$.