Can someone give me an example of a Banach space $X$ and contractive projection $P\in\mathcal{B}(X)$ such that $\ker P$ is not a range of any contractive projection $Q\in\mathcal{B}(X)$?



Contractively complemented subspaces without contractively complemented complement
The projection from the space $c$ of convergent sequences to the $1$-dimensional subspace of constant sequences with kernel $c\_0$. Any projection $c \to c\_0$ has norm at least $2$ (cf. exercises to chapter 2.5 in Albiac-Kalton).


