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Strong and Extremely Strong Ditkin sets forthe Banach Algebras Apr(G) = Ap ⋂ Lr(G)

Published online by Cambridge University Press:  20 November 2018

Edmond E. Granirer
Edmond E. Granirer*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2 email: granirer@math.ubc.ca
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Abstract

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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.

Keywords

43A1543A1046J1043A45Fourier algebraFiga–Talamanca–Herz algebralocally compact groupDitkin setsHelson setsSidon setsweak sequential completeness
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 1 , 01 February 2011 , pp. 123 - 135
Copyright
Copyright © Canadian Mathematical Society 2011

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