I'm reading Robinson & Bratteli's book and, at some point, the following statement is given.
A $C^{*}$-algebra $\mathcal{U}$ is called simple if it has no nontrivial closed two-sided ideals, i.e., if the only closed two-sided ideals are $\{0\}$ and $\mathcal{U}$. If $\mathcal{U}$ has an identity this amounts to saying that $\mathcal{U}$ has no two-sided ideals at all, closed or not.
But why does the second statement hold? I mean, if $\mathcal{U}$ has an identity ${\bf{1}}$, why $\{0\}$ and $\mathcal{U}$ are no longer two-sided ideals? For the first, don't we still have $a0 = 0a = 0 \in \{0\}$, for every $a \in \mathcal{U}$ (and analogously for the second)?