Products of Lipschitz-free spaces and applications

Pedro Levit Kaufmann¹

¹ Instituto de Ciência e Tecnologia Universidade Federal de São Paulo Campus São José dos Campos – Parque Tecnológico Avenida Doutor Altino Bondensan, 500 12247-016 São José dos Campos/SP, Brazil

Studia Mathematica, Tome 226 (2015) no. 3, pp. 213-227

Résumé

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 nonlinear version of Pełczyński'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 includes all $C(K)$ spaces with $K$ infinite compact metric (Dutrieux and Ferenczi (2006) already proved that $\mathcal{F}(C(K))$ is isomorphic to $\mathcal{F}(c_0)$ for those $K$ using a different method).

Détail

DOI : 10.4064/sm226-3-2

Keywords: given banach space lipschitz free space denoted mathcal isomorphic sum infty mathcal ell applications presented including nonlinear version czy skis decomposition method lipschitz free spaces identification isomorphism between mathcal mathbb lipschitz free space compact metric space which locally bi lipschitz embeddable mathbb which contains subset lipschitz equivalent unit ball mathbb mathcal isomorphic mathcal separable metric spaces which absolute lipschitz retracts contain subset which lipschitz equivalent unit ball class includes spaces infinite compact metric dutrieux ferenczi already proved mathcal isomorphic mathcal those using different method