The Daugavet property and translation-invariant subspaces

Simon Lücking¹

¹ Department of Mathematics Freie Universität Berlin Arnimallee 6 14195 Berlin, Germany

Studia Mathematica, Tome 221 (2014) no. 3, pp. 269-291

Résumé

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.

Détail

DOI : 10.4064/sm221-3-5

Keywords: infinite compact abelian group varlambda subset its dual group vargamma study question which spaces form varlambda varlambda which quotients form varlambda varlambda have daugavet property varlambda rich subspace only vargamma setminus varlambda semi riesz set varlambda rich subspace varlambda rich subspace concerning quotients prove varlambda has daugavet property varlambda rosenthal set varlambda poor subspace varlambda nicely placed riesz set