Lipschitz Algebras and Lipschitz-Free Spaces Over Unbounded Metric Spaces

Abstract

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