The almost Daugavet property and translation-invariant subspaces

Abstract: Let $G$ be a metrizable, compact abelian group and let $\Lambda$ be a subset of its dual group $\hat G$. We show that $C_\Lambda(G)$ has the almost Daugavet property if and only if $\Lambda$ is an infinite set, and that $L^1_\Lambda(G)$ has the almost Daugavet property if and only if $\Lambda$ is not a $\Lambda(1)$ set.