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The Daugavet property and translation-invariant subspaces

100%

Lücking S.

Studia Mathematica

2014

tom 221

nr 3

269-291

Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form $C_{Λ}(G)$ or $L¹_{Λ}(G)$ and which quotients of the form $C(G)/C_{Λ}(G)$ or $L¹(G)/L¹_{Λ}(G)$ have the Daugavet property. We show that $C_{Λ}(G)$ is a rich subspace of C(G) if and only if $Γ∖ Λ^{-1}$ is a semi-Riesz set. If $L¹_{Λ}(G)$ is a rich subspace of L¹(G), then $C_{Λ}(G)$ is a rich subspace of C(G) as well. Concerning quotients, we prove that $C(G)/C_{Λ}(G)$ has the Daugavet property if Λ is a Rosenthal set, and that $L¹_{Λ}(G)$ is a poor subspace of L¹(G) if Λ is a nicely placed Riesz set.

The almost Daugavet property and translation-invariant subspaces

100%

Lücking S.

Colloquium Mathematicum

2014

tom 134

nr 2

151-163

Let G be a metrizable, compact abelian group and let Λ be a subset of its dual group Ĝ. We show that $C_{Λ}(G)$ has the almost Daugavet property if and only if Λ is an infinite set, and that $L¹_{Λ}(G)$ has the almost Daugavet property if and only if Λ is not a Λ(1) set.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Studia Mathematica

2 Lücking S.

2 2014

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The Daugavet property and translation-invariant subspaces

The almost Daugavet property and translation-invariant subspaces