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:


1. $\ell^\infty / c\_0$ by Kalton [1, Theorem 4.2].
2. A quotient of a certain Càdlàg type space with $C[0,1]$ by Lindenstrauss and Aharoni [2, Remark ii].


I have two questions:



> 
> 1. Are there any more examples known?
> 2. 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?
> 
> 
> 


[1] Nigel Kalton. Lipschitz and uniform embeddings into ℓ∞. Fund. Math. 212 (2011), no. 1, 53–69. 


[2] Aharoni, Israel; Lindenstrauss, Joram. Uniform equivalence between Banach spaces. Bull. Amer. Math. Soc. 84 (1978), no. 2, 281–283. 



Lipschitz right inverses of Banach space quotients
The answer to (2) is yes and is contained in your reference [2]. That was how Aharoni and Lindenstrauss were able to construct two Lipschitz equivalent Banach spaces that are not isomorphic. Note that their example is non separable. Whether there are two Lipschitz equivalent non isomorphic separable Banach spaces is a famous open problem. Godefroy and Kalton proved that the Aharoni-Lindenstrauss approach cannot work in the separable setting. That is, if a quotient mapping from a separable space has a Lipschitz right inverse, then it has a bounded linear right inverse. This is contained in


Godefroy, G.(F-PARIS6-E); Kalton, N. J.(1-MO)
Lipschitz-free Banach spaces. (English summary) 
Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday.
Studia Math. 159 (2003), no. 1, 121–141.


