Let $X$ be a Banach space and $Y$ a closed subspace of $X$. I am interested in quotients $q:X\to X/Y$ that do not have Lipschitz right inverses (not necessarily linear).
Of course, if $Y$ is complemented, then the quotient always has a Lipschitz (bounded linear) right inverse.
The only examples I know of that do not have Lipschitz right inverses are:
- $\ell^\infty / c_0$ by Kalton [1, Theorem 4.2].
- A quotient of a certain Càdlàg type space with $C[0,1]$ by Lindenstrauss and Aharoni [2, Remark ii].
I have two questions:
- Are there any more examples known?
- Is this perhaps a general feature of non-complemented subspaces of Banach spaces? Or is there an example of a non-complemented subspace $Y$ of some $X$ so that the quotient $q:X\to X/Y$ has a Lipschitz right inverse?
 Nigel Kalton. Lipschitz and uniform embeddings into ℓ∞. Fund. Math. 212 (2011), no. 1, 53–69.
 Aharoni, Israel; Lindenstrauss, Joram. Uniform equivalence between Banach spaces. Bull. Amer. Math. Soc. 84 (1978), no. 2, 281–283.