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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## The Daugavet property and translation-invariant subspaces

### Tom 221 / 2014

Studia Mathematica 221 (2014), 269-291 MSC: Primary 46B04; Secondary 43A46. DOI: 10.4064/sm221-3-5

#### Streszczenie

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.

#### Autorzy

• Simon LückingDepartment of Mathematics
Freie Universität Berlin
Arnimallee 6
14195 Berlin, Germany
e-mail

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