The Daugavet property and translation-invariant subspaces

Simon Lücking. "The Daugavet property and translation-invariant subspaces." Studia Mathematica 221.3 (2014): 269-291. <http://eudml.org/doc/285905>.

@article{SimonLücking2014,
abstract = {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.},
author = {Simon Lücking},
journal = {Studia Mathematica},
keywords = {Daugavet property; nicely placed set; poor subspace; rich subspace; semi-Riesz set; small subspace; translation-invariant subspace},
language = {eng},
number = {3},
pages = {269-291},
title = {The Daugavet property and translation-invariant subspaces},
url = {http://eudml.org/doc/285905},
volume = {221},
year = {2014},
}

TY - JOUR
AU - Simon Lücking
TI - The Daugavet property and translation-invariant subspaces
JO - Studia Mathematica
PY - 2014
VL - 221
IS - 3
SP - 269
EP - 291
AB - 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.
LA - eng
KW - Daugavet property; nicely placed set; poor subspace; rich subspace; semi-Riesz set; small subspace; translation-invariant subspace
UR - http://eudml.org/doc/285905
ER -