Simple $C^*$-algebras are not commutative

Question

Is it true that for simple $C^*$-algebras, meaning that they don't have non-trivial two-sided ideals, it holds that they are necessarily non-commutative? And why?

Answer 1

This is false. $\mathbb C$ is commutative and simple.

But this is the only simple commutative $C^*$-algebra. This is because if $A$ is not isomorphic to $\mathbb C$, then it has a non-zero, non-invertible element $x$ by the Gelfand-Mazur theorem. If $A$ is also commutative, then $Ax$ is a proper ideal.

Answer 2

For a different point of view, a commutative C$^*$-algebra $A$ is isomorphic, via the Gelfand transform, to $C_0(X)$ for a locally compact Hausdorff space $X$. If $X$ is a singleton, then $A=\mathbb C$. Otherwise, for nontrivial closed $Y\subset X$, then set $$J=\{f\in C_0(X):\ f|_J=0\}$$ is a closed two-sided ideal.