splitting lemma in $C^*$ algebras

Question

In abelian category,there is a splitting lemma. see https://en.wikipedia.org/wiki/Splitting_lemma

I wonder whether the splitting lemma also holds in $C^*$ algebras .Is left split equivalent to right split?

Answer 1

No, not even if the C$^*$-algebra is commutative. For instance if you take $A=C_b(\mathbb N)$ (bounded sequences), then $I=c_0(\mathbb N)$ is an ideal that is not complemented.