Open AccessCharacterizations of Riesz sets
Author(s) -
A. Ülger
Publication year - 2011
Publication title -
mathematica scandinavica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.553
H-Index - 30
eISSN - 1903-1807
pISSN - 0025-5521
DOI - 10.7146/math.scand.a-15171
Subject(s) - backslash , mathematics , abelian group , group (periodic table) , combinatorics , measure (data warehouse) , algebra over a field , group algebra , riesz potential , pure mathematics , physics , quantum mechanics , database , computer science
Let $G$ be a compact abelian group, $M(G)$ its measure algebra and $L^{1}(G)$ its group algebra. For a subset $E$ of the dual group $\widehat{G}$, let $M_{E}(G)=\{\mu\in M(G):\widehat{\mu}=0$ on $\widehat{G} \backslash E\}$ and $L_{E}^{1}(G)=\{a\in L^{1}(G):\widehat{a}=0$ on $\widehat{G}\backslash E\}$. The set $E$ is said to be a Riesz set if $M_{E}(G)=L_{E}^{1}(G)$. In this paper we present several characterizations of the Riesz sets in terms of Arens multiplication and in terms of the properties of the Gelfand transform $\Gamma :L_{E}^{1}(G)\rightarrow c_{0}(E)$.