If all unital Banach algebras contain maximal ideals, then why doesn't $M_n(\mathbb{C})$ have proper ideals?

Question

In Murphy's textbook on C*algebra, he writes:

So all unital Banach algebra should contain maximal ideals, but on a StackExchange post, there is an explicit example $M_n(\mathbb{C})$ of an unital (non-commutative) Banach algebra that doesn't even have proper ideals:

Example of a Banach algebra $ A $ whose only closed ideals are $ \{ 0 \} $ and $ A $.

But one should be able to use Zorn's lemma argument to show there exists maximal (and hence proper) ideals so I'm not sure what is going on here. What am I missing?

Answer 1

Zorn's lemma applied to the matrices works just fine. The trivial ideal $\{0\}$ is proper (so you have at least one proper ideal), and a maximal ideal obtained by Zorn's lemma happens to be exactly this very ideal.

A (unital) algebra is simple, if $\{0\}$ is the only proper (hence maximal) ideal thereof. There is a vast literature on simple C*-algebras. Going beyond matrices, you may check the Calkin algebras all bounded operators modulo compact operators, Cuntz algebras, the Jiang-Su algebra etc.