This is a problem in exercise one of Murphy's book
Find an example of a nonabelian unital Banach algebra $A$, where the only closed ideals are $\{0\}$ and $A$.
But does such an algebra exist at all?
My argument is the following:
Let $a$ be an arbitrary nonzero element in $A$, if $a$ is not invertible, then $a$ is contained in a maximal ideal, which is closed. However, there is no such ideal thus every nonzero element must be invertible. Then Gelfand-Mazur says $A$ is the complex numbers and thus must be abelian.
What is the problem with my argument?
Thanks!