Let $A:X\to Y$ be a surjective morphism of Banach spaces.


1) Does there always exists $B\_R$, a bounded right inverse to $A$?


2) Assume additionally that $A$ is a morphism of unital Banach algebras. The same question.



Bounded inverse to morphism of Banach algebras
No, and no. The following is a bit too long to write clearly as a comment.


Let $X=\ell^\infty$. Let $c$ be the unital, closed subalgebra of $X$ that consists of all convergent sequences. Let $Y=X/c$ be the quotient algebra and let $A:X\to Y$ be the quotient homomorphism.


Suppose there exists a bounded linear map $B\_R:Y\to X$ such that $AB\_R$ is the identity map on $Y$. Then $I - B\_RA$ would be a bounded linear projection of $\ell^\infty$ onto $c$. But this is known to be impossible by [Phillips's lemma](https://math.stackexchange.com/questions/132520/complement-of-c-0-in-ell-infty).


