The almost Daugavet property and translation-invariant subspaces

Let $G$ be a metrizable, compact abelian group and let $\mathrm{\Lambda }$ be a subset of its dual group $\widehat{G}$. We show that $C_{\mathrm{\Lambda }}(G)$ has the almost Daugavet property if and only if $\mathrm{\Lambda }$ is an infinite set, and that $L_{\mathrm{\Lambda }}^1(G)$ has the almost Daugavet property if and only if $\mathrm{\Lambda }$ is not a $\mathrm{\Lambda }(1)$ set.