The Daugavet property and translation-invariant subspaces

• Autorzy: Simon Lücking
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46B04: Isometric theory of Banach spaces
• 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2014
• Tom: 221
• Numer: 3
• Strony: 269-291

Daty

wydano

2014

Twórcy

autor

Simon Lücking

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

Identyfikatory

DOI

10.4064/sm221-3-5