The Daugavet property and translation-invariant subspaces
Volume 221 / 2014
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.
