Let $A$ be the Banach algebra of $n \times n$ matrices over $\mathbb C$. Then the subset consisting of all diagonal matrices is an abelian subalgebra. (correct me if I'm wrong).
Now I want to show that it is in fact maximal. (I am quite sure it is but maybe it isn't?).
How can I show that the diagonal matrices are a maximal abelian subalgebra? Of course the way to prove it is to assume that there exists an abelian subalgebra containing the diagonal matrices. But what then?
Prove that if $A$ is a diagonal matrix with pairwise distinct diagonal entries and $AB=BA$, then $B$ is diagonal.