As I asked in this comment here:
If $a$ is an $n \times n$ matrix and we define
$$ \|a\|_p = \left ( \max_j \sum_i |a_{ij}|^p \right)^{1 \over p}$$
or
$$ \|a\|_p = \left ( \max_i \sum_j |a_{ij}|^p \right)^{1 \over p}$$
Does this norm make the set of matrices a Banach algebra?
It's not true at least if $p >\frac{\ln 3}{\ln 2}$. Take the following $2\times 2$ matrices:
$$A = \left(\begin{array}{cc} 1 & 1 \\ 0 & 1\end{array}\right), B = \left(\begin{array}{cc} 1 & 0 \\ 1 & 1\end{array}\right). $$
Then for your first $p$-norm, $||A|| = ||B|| = 2^{\frac{1}{p}}$ and
$$||AB|| = \left| \left(\begin{array}{cc} 2 & 1 \\ 1 & 1\end{array}\right)\right| = (2^p +1)^{\frac{1}{p}} > ||A|| \cdot ||B||. $$
I guess these norms do not define a Banach algebra for all $p \geq 1$. But I cannot find a counterexample yet.
