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

### Tom 136 / 2014

Colloquium Mathematicum 136 (2014), 271-296 MSC: Primary 43A30, 43A45, 43A46; Secondary 22D15, 46J20. DOI: 10.4064/cm136-2-9

#### Streszczenie

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 $\overline {J(E)}^{w^{\ast }}=B(G)$).

#### Autorzy

• A. ÜlgerDepartment of Mathematics
Koc University
34450 Sariyer, Istanbul, Turkey
e-mail

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