On large ℓ₁-sums of Lipschitz-free spaces and applications

Abstract

We prove that the Lipschitz-free space over a Banach space X X of density κ \kappa , denoted by F ( X ) \mathcal {F}(X) , is linearly isomorphic to its ℓ 1 \ell _1 -sum ( ⨁ κ F ( X ) ) ℓ 1 \left (\bigoplus _{\kappa }\mathcal {F}(X)\right )_{\ell _1} . This provides an extension of a previous result from Kaufmann in the context of non-separable Banach spaces. Further, we obtain a complete classification of the spaces of real-valued Lipschitz functions that vanish at 0 0 over a L p \mathcal {L}_p -space. More precisely, we establish that, for every 1 ≤ p ≤ ∞ 1\leq p\leq \infty , if X X is a L p \mathcal {L}_p -space of density κ \kappa , then L i p 0 ( X ) \mathrm {Lip}_0(X) is either isomorphic to L i p 0 ( ℓ p ( κ ) ) \mathrm {Lip}_0(\ell _p(\kappa )) if p > ∞ p>\infty , or L i p 0 ( c 0 ( κ ) ) \mathrm {Lip}_0(c_0(\kappa )) if p = ∞ p=\infty .