# Banach-Stone Theorem in Lipschitz-free spaces

If $T$ is a nonlinear surjective isometry from Lipschitz-free space $\mathcal{F}(M)$ to $\mathcal{F}(N)$($M,N$ are metric spaces), is $M$ homeomorphic to $N$?

• I might be missing something but non-linearity doesn't seem to be essential here. Indeed, by the Mazur-Ulam theorem such map $T$ is affine and hence $T-T(0)I$ is a linear isometry. May 1, 2015 at 8:23
• Aha!I am missing the Mazur-Ulam theorem.Thank you. May 1, 2015 at 9:06
• Another question is : if $T$ is a Lipschitz isomorphism, is $M$ homeomorphic to $N$? May 1, 2015 at 9:09
• To Dongyang Chen: It looks like Weaver has answered your question. You did not accept the answer, does this mean that you find it incomplete? Jan 24, 2016 at 19:23

No, this is false --- trivially, because if $$M$$ is the completion of $$N$$ then $$M$$ and $$N$$ have the same Arens-Eells space. But if you require $$M$$ and $$N$$ to both be complete there is a more interesting counterexample.
Let $$M$$ be three copies of the interval $$[0,1]$$, joined at the $$0$$'s, with path metric. (That is, a capital "Y" with path metric.) Let $$N$$ just be a single copy of the interval $$[0,3]$$. Then $${\rm Lip}_0(M)$$ and $${\rm Lip}_0(N)$$ are, respectively, isometrically isomorphic to $$L^\infty(M)$$ and $$L^\infty(N)$$, and are therefore isometrically isomorphic to each other, by cutting $$N$$ into three pieces and matching up. (To go from $$L^\infty(M)$$ to $${\rm Lip}_0(M)$$, integrate from the origin.) The Arens-Eells spaces of $$M$$ and $$N$$ sit inside of the dual spaces $${\rm Lip}_0(M)^*$$ and $${\rm Lip}_0(N)^*$$ and it is straightforward to check that the dual isometric isomorphism between these spaces restricts to an isometric isomorphism of $$AE(M)$$ and $$AE(N)$$.
(If you work through this example you will see how to define the map between $$AE(M)$$ and $$AE(N)$$ directly, but I think the explanation I gave above is more illuminating as to why this isometric isomorphism exists.)
On the positive side, there are some good Banach-Stone type theorems if you impose additional conditions on $$M$$ and $$N$$; see Sections 3.7 and 3.8 of the second edition of my book on Lipschitz Algebras.
Edit: The "capital Y" example I described in this answer is not new. A. Godard (Tree metrics and their Lipschitz-free spaces, Proc. AMS 138 (2010), 4311-4320) proves that the Arens-Eells space of any separable metric tree is isometrically isomorphic to $$L^1[0,1]$$. Second comment: If $$M$$ and $$N$$ are compact and concave in the sense of this question and $$AE(M)$$ is isometrically isomorphic to $$AE(N)$$, then $$M$$ and $$N$$ are isometric up to a dilation. (Compactness is not needed if one assumes a uniform version of concavity.) This is Theorem 3.55 of the second edition of my book.