This is an announcement for the paper "Products of Lipschitz-free spaces and applications" by Pedro Levit Kaufmann.
Abstract: We show that, given a Banach space $X$, the Lipschitz-free space over $X$, denoted by $\mathcal{F}(X)$, is isomorphic to $(\sum_{n=1}^\infty \mathcal{F}(X))_{\ell_1}$. Some applications are presented, including a non-linear version of Pe\l czy'ski's decomposition method for Lipschitz-free spaces and the identification up to isomorphism between $\mathcal{F}(\mathbb{R}^n)$ and the Lipschitz-free space over any compact metric space which is locally bi-Lipschitz embeddable into $\mathbb{R}^n$ and which contains a subset that is Lipschitz equivalent to the unit ball of $\mathbb{R}^n$. We also show that $\mathcal{F}(M)$ is isomorphic to $\mathcal{F}(c_0)$ for all separable metric spaces $M$ which are absolute Lipschitz retracts and contain a subset which is Lipschitz equivalent to the unit ball of $c_0$. This class contains all $C(K)$ spaces with $K$ infinite compact metric (Dutrieux and Ferenczi had already proved that $\mathcal{F}(C(K))$ is isomorphic to $\mathcal{F}(c_0)$ for those $K$ using a different method). Finally we study Lipschitz-free spaces over certain unions and quotients of metric spaces, extending a result by Godard.
Archive classification: math.FA
Mathematics Subject Classification: Primary 46B20, Secondary 46T99
Remarks: 17 pages, 1 figure
Submitted from: pkaufmann@math.jussieu.fr
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1403.6605
or