Compatifiant Algebras and the C*-algebras of the affine group

Let G be the affine group. Let π be one of the infinite representation of G, acting on L ² (ℝ ⁺). It is known that G is not liminar. However, there are bounded operators A on L ² (ℝ ⁺) such that the operators Aπ(ƒ) and π(ƒ)A are compact, ∀ƒ ∊ L¹(G). In this paper, we prove that these compactness conditions are not sufficient to characterize the C*.algebras of G. For that, we introduce the new notions of compactifiant algebras and selbicompactifiant algebras. We present some properties of these algebras, showing similarities with Von Newmann algebras. We determine all bounded functions ϕ such that the operator M_ϕπ(ƒ) and (ƒ)M_ϕ are compact, ∀ƒ ∊ L¹ (G), where M_ϕ is the product operator by ϕ.