A lifting theorem for operators between Lipschitz spaces

We prove that every bounded linear operator between Lipschitz spaces admits a lifting along the de Leeuw embedding. More precisely, given pointed metric spaces $M$ and $N$ and $\epsilon >0$, every bounded linear operator $S:{\mathrm{L}\mathrm{i}\mathrm{p}}_0(M)\to {\mathrm{L}\mathrm{i}\mathrm{p}}_0(N)$ admits a lifting $\mathfrak{S}:C(\beta \tilde{M})\to C(\beta \tilde{N})$ such that $\mathrm{\parallel }\mathfrak{S}\mathrm{\parallel }\le \mathrm{\parallel }S\mathrm{\parallel }+\epsilon$ and $\mathfrak{S}({\Phi }_M(f))={\Phi }_N(S(f))$ for every $f\in {\mathrm{L}\mathrm{i}\mathrm{p}}_0(M)$. Moreover, compact operators admit compact liftings.