Relatively weak$^{\ast }$ closed ideals of $A(G)$, sets of synthesis and sets of uniqueness

A. Ülger¹

¹ Department of Mathematics Koc University 34450 Sariyer, Istanbul, Turkey

Colloquium Mathematicum, Tome 136 (2014) no. 2, pp. 271-296.

Résumé

Let $G$ be a locally compact amenable group, and $A(G)$ and $B(G)$ the Fourier and Fourier–Stieltjes algebras of $G$. For a closed subset $E$ of $G$, let $J(E)$ and $k(E)$ be the smallest and largest closed ideals of $A(G)$ with hull $E$, respectively. We study sets $E$ for which the ideals $J(E)$ or/and $k(E)$ are $\sigma (A(G),C^{\ast }(G))$-closed in $A(G)$. Moreover, we present, in terms of the uniform topology of $C_{0}(G)$ and the weak$^{\ast }$ topology of $B(G)$, a series of characterizations of sets obeying synthesis. Finally, closely related to the above issues, we present a series of results about closed sets of uniqueness (i.e. closed sets $E$ for which $\overline {J(E)}^{w^{\ast }}=B(G)$).

Détail

DOI : 10.4064/cm136-2-9

Keywords: locally compact amenable group fourier fourier stieltjes algebras closed subset smallest largest closed ideals hull respectively study sets which ideals sigma ast closed moreover present terms uniform topology weak ast topology series characterizations sets obeying synthesis finally closely related above issues present series results about closed sets uniqueness closed sets which overline ast