Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces (Q6110800)

The paper focuses on the natural question on how the ultraproducts of metric spaces and ultraproducts of Lipschitz-free Banach spaces over such metric spaces are related. It provides fairly comprehensive answers, some of which are listed below. \begin{itemize} \item[1.] {Embeddability of Lipschitz-free spaces over ultraproducts of metric spaces into ultraproducts of Lipschitz-free spaces:} If \(I\) is a set of indices, \((M_i)_{i\in I}\) is a set of (pointed) metric spaces indexed by \(I\), and \(\mathcal{U}\) is an ultrafilter over \(I\), then \(\mathcal{F}((M_i)_\mathcal{U})\), the Lipschitz-free space over the \(\mathcal{U}\)-ultraproduct of \((M_i)_i\), is linearly isometric to the closed linear span of the set \(\{\delta(x)\colon x\in (M_i)_\mathcal{U}\}\) inside the Banach space ultraproduct \(\big(\mathcal{F}(M_i)\big)_\mathcal{U}\), where the former set is embedded into the latter via the canonical embedding. This has as immediate consequence (using also the fact that an ultrapower of a Banach space is finitely representable in itself) that, for any metric space \(M\) and an ultrafilter \(\mathcal{U}\) over some set, \(\mathcal{F}(M_\mathcal{U})\) is finitely representable in \(\mathcal{F}(M)\), where \(M_\mathcal{U}\) is the \(\mathcal{U}\)-ultrapower of \(M\). \item[2.] {Implications of coarse embeddability between metric spaces to finite representability between the corresponding Lipschitz-free spaces:} If \(X\) and \(Y\) are Banach spaces such that \(X\) coarsely embeds into \(Y\), then \(\mathcal{F}(X)\) is crudely finitely representable in \(\mathcal{F}(Y)\). \item[3.] {Relations between metric type/cotype and Rademacher cotype of the corresponding Lipschitz-free space:} If \(M\) is a metric space such that the Banach space \(\mathcal{F}(M)\) has Rademacher cotype, then \(M\) has metric type in the sense of Bourgain, Milman, Wolfson [\textit{J.~Bourgain} et al., Trans. Am. Math. Soc. 294, 295--317 (1986; Zbl 0617.46024)]. \item[4.] {Stability of Lipschitz-free spaces under ultraproducts:} Let \(\mathcal{U}\) be a countably-incomplete ultrafilter over an infinite set, in particular, any non-principal ultrafilter over a countably infinite set, and let \(M\) and \(X\) be a metric space, resp. a Banach space. Then \(X_\mathcal{U}\) is not isomorphic to a subspace of \(\mathcal{F}(M)\). In particular, an ultraproduct of Lipschitz-free spaces does not embed into a Lipschitz-free space. \end{itemize}