Weighted measure algebras and uniform norms
Volume 177 / 2006
Studia Mathematica 177 (2006), 133-139
MSC: 43A10, 43A20, 46J05.
DOI: 10.4064/sm177-2-3
Abstract
Let $\omega$ be a weight on an LCA group $G$. Let ${M(G, \omega)}$ consist of the Radon measures $\mu$ on $G$ such that $\omega\mu$ is a regular complex Borel measure on $G$. It is proved that: (i) ${M(G, \omega)}$ is regular iff ${M(G, \omega)}$ has unique uniform norm property (UUNP) iff ${L^1(G, \omega)}$ has UUNP and $G$ is discrete; (ii) ${M(G, \omega)}$ has a minimum uniform norm iff ${L^1(G, \omega)}$ has UUNP; (iii) ${M_{00}(G, \omega)}$ is regular iff ${M_{00}(G, \omega)}$ has UUNP iff ${L^1(G, \omega)}$ has UUNP, where ${M_{00}(G, \omega)} := \{\mu \in {M(G, \omega)} : \widehat\mu = 0 \hbox{ on } {\mit\Delta} ({M(G, \omega)}) \setminus {\mit\Delta} ({L^1(G, \omega)}) \}$.
