Bounded inverse to morphism of Banach algebras

Question

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.

Answer 1

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.

Answer 2

For ${\cal H}$ an infinite-dimensional Hilbert space, the canonical projection $\cal{B(H)}\rightarrow \cal{A(H)}$ (where $\cal{A(H)}$ is the Calkin algebra), has no continuous, linear section; this is due to E.O. Thorp, Link