Convex integrals of molecules in Lipschitz-free spaces (Q6585643)

The setting is that \((M,d,0)\) is a complete pointed metric space, \(\operatorname{Lip}_0(M)\) the Banach space of real-valued Lipschitz functions and \(\mathcal{F}(M)\) its canonical predual known as \textit{the Lipschitz-free space over \(M\)}.\N\NBy \(\widetilde{M}\) we denote the set \(M\times M\setminus\{(x,x)\colon x\in M\}\). The \textit{de Leeuw transform} is the linear isometry \(\Phi:\operatorname{Lip}_0(M)\to C(\beta\widetilde{M})\) given by the unique continuous extension of the mapping\N\[\N\Phi(f)(x,y) = \frac{f(x)-f(y)}{d(x,y)},\quad (x,y)\in \widetilde{M}.\N\]\NIt is known that then \(\operatorname{Lip}_0(M)^* = \Phi^*(\mathcal{M}(\beta\widetilde{M}))\), where \(\mathcal{M}(\beta\widetilde{M}) = C(\beta\widetilde{M})^*\) is the space of Radon measures. The purpose of this paper is to initiate a systematic study of points in \(\operatorname{Lip}_0(M)^*\) (and its subspace \(\mathcal{F}(M)\)) thinking of those as images of some \(\mu\in\mathcal{M}(\beta\widetilde{M})\) under the mapping \(\Phi^*\).\N\NIn Proposition 2.6 the authors observe that for any \(\mu\in \mathcal{M}(\beta\widetilde{M})\) we have\N\[\N\Phi^*(\mu|_{\widetilde{M}}) = \int_{\widetilde{M}} m_{x,y} \, d\mu(x,y)\N\]\Nas a Bochner integral in \(\mathcal{F}(M)\), where the \(m_{x,y}\) are normalized elementary molecules. Based on this observation we say that \(\gamma\in\mathcal{F}(M)\) is a \textit{convex integral of molecules} if there exists \(\mu\in \mathcal{M}(\beta\widetilde{M})\) concentrated on \(\widetilde{M}\) such that \(\mu\geq 0\), \(\Phi^*(\mu)=\gamma\) and \(\|\gamma\| = \|\mu\|\); if the measure \(\mu\) is discrete we say that \(\gamma\) is a \textit{convex series of molecules}.\N\NThe authors systematically study which points in a Lipschitz-free space are convex integrals/series of molecules, their additional features, give various examples witnessing complexity of the notion and finally, they give an application of it. An example of results obtained by the authors are: if \(M\) is a uniformly discrete space, then any point in \(\mathcal{F}(M)\) is a convex series of molecules (see Corollary~3.7) and any extreme point in \(\mathcal{F}(M)\) is an elementary molecule (see Corollary~6.3).\N\NThis paper has at least two important follow-ups written by the same authors. The first one is the preprint [\textit{R.~J. Aliaga} et al., ``De Leeuw representations of functionals on Lipschitz spaces'', Preprint, \url{arXiv:2403.09546}], where the authors e.g. extend the content of Section~2 to a more general context. The second is the very recent preprint [\textit{R.~J. Aliaga} et al., ``A solution to the extreme point problem and other applications of Choquet theory to Lipschitz-free spaces'', Preprint, \url{arXiv:2412.04312}], where the authors e.g. prove that any extreme point in \(\mathcal{F}(M)\) is an elementary molecule (extending the above mentioned Corollary~6.3) or that whenever \(M\) is scattered, then any point in \(\mathcal{F}(M)\) is a convex series of molecules (extending the above mentioned Corollary~3.7).