Banach algebra norms on $M_n(A)$

Question

Let $A$ be a Banach algebra (not sure whether I need $A$ to be unital). I saw the claim that all Banach algebra norms on $M_n(A)$ with continuous projections on entries are equivalent. How does one prove this?

Answer 1

The assumption is that the map $X\to X_{kj}$ is continuous for each $k,j$. This means that there exists a constant $c$ such that $\|X_{kj}\|\leq c\|X\|$ for all $X\in M_n(A)$. Consider, on $M_n(A)$, the norm $$ \|X\|_1=\sum_{k,j}\|X_{kj}\|. $$ Note that $(M_n(A),\|\cdot\|_1)$ is a Banach space. The above shows that $\|X\|_1\leq cn^2\|X\|$, so the identity map is continuous $(M_n(A),\|\cdot\|)\to(M_n(A),\|\cdot\|_1)$. As it is bijective, the Open Mapping Theorem guarantees that there is a constant $d$ with $\|X\|\leq d\,\|X\|_1$ for all $X\in M_n(A)$.