Products of Lipschitz-free spaces and applications

Pedro Levit Kaufmann. "Products of Lipschitz-free spaces and applications." Studia Mathematica 226.3 (2015): 213-227. <http://eudml.org/doc/285767>.

@article{PedroLevitKaufmann2015,
abstract = {We show that, given a Banach space X, the Lipschitz-free space over X, denoted by ℱ(X), is isomorphic to $(∑_\{n=1\}^\{∞\}ℱ(X))_\{ℓ₁\}$. Some applications are presented, including a nonlinear version of Pełczyński’s decomposition method for Lipschitz-free spaces and the identification up to isomorphism between ℱ(ℝⁿ) and the Lipschitz-free space over any compact metric space which is locally bi-Lipschitz embeddable into ℝⁿ and which contains a subset that is Lipschitz equivalent to the unit ball of ℝⁿ. We also show that ℱ(M) is isomorphic to ℱ(c₀) for all separable metric spaces M which are absolute Lipschitz retracts and contain a subset which is Lipschitz equivalent to the unit ball of c₀. This class includes all C(K) spaces with K infinite compact metric (Dutrieux and Ferenczi (2006) already proved that ℱ(C(K)) is isomorphic to ℱ(c₀) for those K using a different method).},
author = {Pedro Levit Kaufmann},
journal = {Studia Mathematica},
keywords = {Lipschitz free spaces; nonlinear Pełczyński decomposition; geometry of Banach spaces; spaces of Lipschitz functions},
language = {eng},
number = {3},
pages = {213-227},
title = {Products of Lipschitz-free spaces and applications},
url = {http://eudml.org/doc/285767},
volume = {226},
year = {2015},
}

TY - JOUR
AU - Pedro Levit Kaufmann
TI - Products of Lipschitz-free spaces and applications
JO - Studia Mathematica
PY - 2015
VL - 226
IS - 3
SP - 213
EP - 227
AB - We show that, given a Banach space X, the Lipschitz-free space over X, denoted by ℱ(X), is isomorphic to $(∑_{n=1}^{∞}ℱ(X))_{ℓ₁}$. Some applications are presented, including a nonlinear version of Pełczyński’s decomposition method for Lipschitz-free spaces and the identification up to isomorphism between ℱ(ℝⁿ) and the Lipschitz-free space over any compact metric space which is locally bi-Lipschitz embeddable into ℝⁿ and which contains a subset that is Lipschitz equivalent to the unit ball of ℝⁿ. We also show that ℱ(M) is isomorphic to ℱ(c₀) for all separable metric spaces M which are absolute Lipschitz retracts and contain a subset which is Lipschitz equivalent to the unit ball of c₀. This class includes all C(K) spaces with K infinite compact metric (Dutrieux and Ferenczi (2006) already proved that ℱ(C(K)) is isomorphic to ℱ(c₀) for those K using a different method).
LA - eng
KW - Lipschitz free spaces; nonlinear Pełczyński decomposition; geometry of Banach spaces; spaces of Lipschitz functions
UR - http://eudml.org/doc/285767
ER -