*-regularity of certain generalized group algebras

Abstract

Let B be a *-semisimple Banach algebra with a bounded approximate identity and ${\alpha: G \longrightarrow {\rm Aut}_{*}(B)}$ (isometric *-automorphisms group of B) an action of a locally group G on B. Let (D, G, γ) be the associated dynamical system, where D = C ₀(G, B) is the Banach *-algebra of all continuous B-valued functions on G vanishing at infinity and the action γ : G → Aut D is given by γ _s (y)(t) = α _s (y(s ⁻¹ t)) for ${y \in D}$ and ${s, t \in G}$ . Recall that B is said to be *-regular if the natural mapping ${I\in {\rm Prim} \, C^{*}(B) \mapsto I\cap B\in {\rm Prim}_{*}(B)}$ is a homeomorphism under the hull-kernel topology. When G is amenable, we show that if B is *-regular, then the generalized group algebra L ¹(G, D; γ) is *-regular. The converse is also true if we further assume that G is countable discrete. Finally the case of compact groups is studied.