Lipschitz-free spaces over products of sequences

Abstract

We answer positively a question of Aliaga and show that for any nonconstant real polynomial $p$, the Lipschitz-free space over $\{(p(n), p(m)):n, m\in \mathbb{N}\}$ is isomorphic to $\mathcal{F}(\mathbb{Z}^2)$. We in fact show more generally that if $d\in \mathbb{N}$, $q\in \mathbb{Z}_{\geq 0}$, and $((a_n^{(i)})_{n=1}^\infty)_{i=1}^d$, $((b_m^{(j)})_{m=1}^\infty)_{j=1}^q$ are sequences with $0 1$ for all $i, j$, then the Lipschitz-free space over the product of these $d+q$ sequences is isomorphic to $\mathcal{F}(\mathbb{Z}^d)$.