Model theory of convolution algebras

Abstract

This paper deals with the model theory of convolution algebras $(L^1(G),*)$ for locally compact groups $G$, seen as Banach lattices equipped with the convolution product $*$. We first prove transfer principles for elementary equivalence and elementary embeddings when the underlying group $G$ is discrete, namely, $(\ell^1(G),*) \equiv (\ell^1(H),*)$ implies $G \equiv H$, while the converse holds when $G$ and $H$ are $\omega$-saturated (likewise for elementary substructures). Without $\omega$-saturation, the converse fails. Although pure Banach lattices are model-theoretically tame, our results imply that adding convolution yields wild behavior. For example, we prove that if $G$ is any locally compact, non-discrete group, then the formula $d(x,x*y)$ is unstable with respect to $\mathrm{Th}(L^1(G),*)$. Moreover, we show that if $G$ is discrete and contains a particular configuration of amenable subgroups, then the formula $d(x*y,z)\mathbin{\dot{-}}\frac{1}{2}$ witnesses $\mathrm{TP}_2$ with respect to $\mathrm{Th}(\ell^1(G),*)$. As a consequence, if $G$ contains an infinite abelian subgroup, then $\mathrm{Th}(\ell^{1}(G),*)$ has $\mathrm{TP}_{2}$. We prove similar results in the locally compact non-discrete setting using the notion of an approximate identity. Finally, we prove a `continuous-by-discrete' approximation theorem. Namely, convolution algebras of connected abelian Lie groups admit metric embeddings into ultraproducts of convolution algebras over finite abelian groups.