On large $\ell_1$-sums of Lipschitz-free spaces and applications

Abstract

We prove that the Lipschitz-free space over a Banach space $X$ of density $\kappa$, denoted by $\mathcal{F}(X)$, is linearly isomorphic to its $\ell_1$-sum $\left(\bigoplus_{\kappa}\mathcal{F}(X)\right)_{\ell_1}$. 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 $0$ over a $\mathcal{L}_p$-space. More precisely, we establish that, for every $1\leq p\leq \infty$, if $X$ is a $\mathcal{L}_p$-space of density $\kappa$, then $\mathrm{Lip}_0(X)$ is either isomorphic to $\mathrm{Lip}_0(\ell_p(\kappa))$ if $p<\infty$, or $\mathrm{Lip}_0(c_0(\kappa))$ if $p=\infty$.