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Strong and Extremely Strong Ditkin sets for the Banach Algebras $A_p^r(G)=A_p\cap L^r(G)$

  Published:2010-11-06
 Printed: Feb 2011
  • Edmond E. Granirer,
    Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2
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Abstract

Let $A_p(G)$ be the Figa-Talamanca, Herz Banach Algebra on $G$; thus $A_2(G)$ is the Fourier algebra. Strong Ditkin (SD) and Extremely Strong Ditkin (ESD) sets for the Banach algebras $A_p^r(G)$ are investigated for abelian and nonabelian locally compact groups $G$. It is shown that SD and ESD sets for $A_p(G)$ remain SD and ESD sets for $A_p^r(G)$, with strict inclusion for ESD sets. The case for the strict inclusion of SD sets is left open. A result on the weak sequential completeness of $A_2(F)$ for ESD sets $F$ is proved and used to show that Varopoulos, Helson, and Sidon sets are not ESD sets for $A_2(G)$, yet they are such for $A_2^r(G)$ 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 SD or ESD sets for $\sigma$-compact groups. The above results are new even if $G$ is abelian.
Keywords: Fourier algebra, Figa-Talamanca-Herz algebra, locally compact group, Ditkin sets, Helson sets, Sidon sets, weak sequential completeness Fourier algebra, Figa-Talamanca-Herz algebra, locally compact group, Ditkin sets, Helson sets, Sidon sets, weak sequential completeness
MSC Classifications: 43A15, 43A10, 46J10, 43A45 show english descriptions $L^p$-spaces and other function spaces on groups, semigroups, etc.
Measure algebras on groups, semigroups, etc.
Banach algebras of continuous functions, function algebras [See also 46E25]
Spectral synthesis on groups, semigroups, etc.
43A15 - $L^p$-spaces and other function spaces on groups, semigroups, etc.
43A10 - Measure algebras on groups, semigroups, etc.
46J10 - Banach algebras of continuous functions, function algebras [See also 46E25]
43A45 - Spectral synthesis on groups, semigroups, etc.
 

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