The Daugavet property and translation-invariant subspaces

Abstract

Let $G$ be an infinite, compact abelian group and let $\varLambda$ be a subset of its dual group $\varGamma$. We study the question which spaces of the form $C_\varLambda(G)$ or $L^1_\varLambda(G)$ and which quotients of the form $C(G)/C_\varLambda(G)$ or $L^1(G)/L^1_\varLambda(G)$ have the Daugavet property. We show that $C_\varLambda(G)$ is a rich subspace of $C(G)$ if and only if $\varGamma \setminus \varLambda^{-1}$ is a semi-Riesz set. If $L^1_\varLambda(G)$ is a rich subspace of $L^1(G)$, then $C_\varLambda(G)$ is a rich subspace of $C(G)$ as well. Concerning quotients, we prove that $C(G)/C_\varLambda(G)$ has the Daugavet property, if $\varLambda$ is a Rosenthal set, and that $L^1_\varLambda(G)$ is a poor subspace of $L^1(G)$, if $\varLambda$ is a nicely placed Riesz set.