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


Published:20101106
Printed: Feb 2011
Edmond E. Granirer,
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2
Abstract
Let $A_p(G)$ be the FigaTalamanca,
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, FigaTalamancaHerz algebra, locally compact group, Ditkin sets, Helson sets, Sidon sets, weak sequential completeness
Fourier algebra, FigaTalamancaHerz 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.
