Quelques observations concernant les ensembles de Ditkin d'un groupe localement compact

On certain products of Banach algebras with applications to harmonic analysis

“…For example, in the context of the Fourier algebra A(G), it has been studied by Derighetti, Forrest, Kaniuth, Lau and the author [6,10,16,22]. In particular it is known that A(G) (or in general, A p (G)) satisfies Ditkin's condition if G is amenable [22, Proposition 3.9, p. 420].…”

Invariant projections and convolution operators

“…For H normal in G, this corollary is already in [3], p. 103. Our proof there was completely different: it was based on the use of A p (G/H).…”

“…The existence of a P with these properties is new even for G abelian and p = 2; if p = 2 and G is abelian the result is due to C. Herz [6]. As a main application we prove that every closed neutral subgroup is a set of p-synthesis in G and locally p-Ditkin in G. A closed subset F of G is locally p-Ditkin in G (see [3], p. 102) if for every ε > 0 and every u ∈ A p (G) ∩ C 00 (G) with Res F u = 0 there is v ∈ A p (G) ∩ C 00 (G) with supp v ∩ F = ∅ and ||u − uv|| Ap(G) < ε. F is p-Ditkin in G if for every ε > 0 and every u ∈ A p (G) with Res F u = 0 there is v ∈ A p (G) ∩ C 00 (G) with supp v ∩ F = ∅ and ||u − uv|| Ap(G) < ε. The method used in the construction of P gives the following extension theorem concerning the Figà-Talamanca Herz algebra of G:…”

Spectral synthesis for $A(G)$ and subspaces of $VN(G)$

“…To begin with, we recall from [3,5,11,14] the local variants of the notions of set of synthesis and Ditkin set. A closed subset E of G is said to be a local spectral set or set of local synthesis if…”

“…Applying Theorem 3.4, Lemma 3.2 and Lemma 2.5, we obtain, in particular, injection theorems for spectral sets and local spectral sets. The injection theorem for local spectral sets has previously been shown in [5,Proposition 8], whereas the injection theorem for spectral sets appears to be new.…”