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

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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?

Bill
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    In $M_n(\mathbb{C})$ the only maximal ideal is $0$. A maximal ideal need not be non-trivial. The complex numbers ($n=1$) the simplest example of such a C*-algebra. – Thusle Gadelankz Aug 04 '23 at 12:27
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    Note that your text just says that a maximal ideal has to be proper, but it does not say it should be non-zero. – J. De Ro Aug 04 '23 at 13:20

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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.

Tomasz Kania
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