The Banach spaces of scalar-valued Lipschitz functions over metric spaces are a frequently studied class of Banach spaces, largely due to the importance of their preduals, the Lipschitz-free spaces. As long as the underlying metric space is infinite, these spaces are non-separable, in fact, they contain an isomorphic copy of $\ell_\infty$, making it natural to search for smaller, more manageable subspaces.

In this talk, I will present my work in progress on $\text{weak}^*$-closed subspaces of spaces of Lipschitz functions over infinite graphs that are invariant under (certain) graph automorphisms. I will provide several motivations for this research and describe its connections to linear cellular automata. Finally, I will use abstract harmonic analysis to prove the following dichotomy: for the $d$-dimensional infinite grid $\mathbb{Z}^d$, every $\text{weak}^*$-closed translation-invariant subspace is either finite-dimensional or contains an isomorphic copy of $\ell_\infty$.