Lipschitz Algebras and Lipschitz-Free Spaces Over Unbounded Metric Spaces

We present a way to turn an arbitrary (unbounded) metric space $\mathcal{M}$ into bounded $\mathcal{B}$ in such that the corresponding Lipschitz-free spaces $\mathcal{F}(\mathcal{M})$ and $\mathcal{F}(\mathcal{B})$ are isomorphic. The construction we provide is functorial weak sense has advantage of being explicit. Apart from its intrinsic theoretical interest, it many applications allows transfer arguments valid for over unbounded spaces. Furthermore, show with slightly modified point-wise multiplication, $\rm{Lip}_0(\mathcal{M})$ scalar-valued Lipschitz functions vanishing at zero any pointed Banach algebra canonical norm.