Quelques observations concernant les ensembles de Ditkin d'un groupe localement compact. (Ditkin sets on locally compact groups) (Q1070161)

We prove that every closed normal subgroup H of a locally compact amenable group G is a Ditkin set with respect to the Herz Figà- Talamanca algebra \(A_ p(G)\) \((p>1)\). We study the injectivity property of Ditkin sets. Let \(\omega\) be the canonical map of G onto G/H and F a closed subset of G/H. We show that F is a Ditkin set if and only if every \(u\in A_ p(G)\) which vanishes on \(\omega^{-1}(F)\), lies in the norm closure of the subspace of \(A_ p(G)\) generated by \(\{\) v \(u_ h|\) \(h\in H\), \(v\in A_ p(G)\cap C_{\infty}(G)\), supp \(v\cap \omega^{- 1}(F)=\emptyset \}\) where \(u_ h(x)=u(xh)\). As far as we know, this result seems to be new even for G abelian and \(p=2\). Weaker formulations are obtained for non amenable groups. We also get a noncommutative version of the Reiter relativization theorems of the spectrum.