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arXiv · 2011.12993

Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces

Fernando Albiac·Jose L. Ansorena·Marek Cuth·Michal Doucha
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Abstract

We present a way to turn an arbitrary (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 point-wise multiplication, the space $\rm{Lip}_0(\mathcal{M})$ of scalar-valued Lipschitz functions vanishing at zero over any (unbounded) pointed metric space is a Banach algebra with its canonical Lipschitz norm.

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Fernando Albiac, Jose L. Ansorena, Marek Cuth, Michal Doucha. 2020-12-05. Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces. https://doi.org/10.1093/imrn%2Frnab193

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