The almost Daugavet property and translation-invariant subspaces

TL;DR

This paper characterizes when certain translation-invariant subspaces of functions on a compact abelian group exhibit the almost Daugavet property, linking it to properties of the subset of the dual group.

Contribution

It provides a complete characterization of the almost Daugavet property for these subspaces based on the nature of the subset of the dual group.

Findings

$C_ ext{Lambda}(G)$ has the almost Daugavet property iff $ ext{Lambda}$ is infinite.

$L^1_ ext{Lambda}(G)$ has the almost Daugavet property iff $ ext{Lambda}$ is not a $ ext{Lambda}(1)$ set.

The results connect harmonic analysis properties with geometric Banach space properties.

Abstract

Let $G$ be a metrizable, compact abelian group and let $\Lambda$ be a subset of its dual group $\hat G$. We show that $C_\Lambda(G)$ has the almost Daugavet property if and only if $\Lambda$ is an infinite set, and that $L^1_\Lambda(G)$ has the almost Daugavet property if and only if $\Lambda$ is not a $\Lambda(1)$ set.