Strong and Extremely Strong Ditkin sets forthe Banach Algebras Apr (G) = A p ⋂ L r (G)

Granirer, Edmond E.

Canadian journal of mathematics, Tome 63 (2011) no. 1, pp. 123-135

Résumé

Let ${{A}_{p}}\left( G \right)$ be the Figa-Talamanca, Herz Banach Algebra on $G$ ; thus ${{A}_{2}}\left( G \right)$ is the Fourier algebra. Strong Ditkin $\left( \text{SD} \right)$ and Extremely Strong Ditkin $\left( \text{ESD} \right)$ sets for the Banach algebras $A_{P}^{r}\left( G \right)$ are investigated for abelian and nonabelian locally compact groups $G$ . It is shown that $\text{SD}$ and $\text{ESD}$ sets for ${{A}_{p}}\left( G \right)$ remain $\text{SD}$ and $\text{ESD}$ sets for $A_{P}^{r}\left( G \right)$ , with strict inclusion for $\text{ESD}$ sets. The case for the strict inclusion of $\text{SD}$ sets is left open.A result on the weak sequential completeness of ${{A}_{2}}\left( F \right)$ for $\text{ESD}$ sets $F$ is proved and used to show that Varopoulos, Helson, and Sidon sets are not $\text{ESD}$ sets for ${{A}_{2}}\left( G \right)$ , yet they are such for $A_{2}^{r}\left( G \right)$ for discrete groups $G$ , for any $1\,\le \,r\,\le \,2$ .A result is given on the equivalence of the sequential and the net definitions of $\text{SD}$ or $\text{ESD}$ sets for $\sigma $ -compact groups.The above results are new even if $G$ is abelian.

Détail

DOI : 10.4153/CJM-2010-077-0

Mots-clés : 43A15, 43A10, 46J10, 43A45, Fourier algebra, Figa–Talamanca–Herz algebra, locally compact group, Ditkin sets, Helson sets, Sidon sets, weak sequential completeness