*-regularity of certain generalized group algebras

Abstract

Let B be a * -semisimple Banach algebra with a bounded approximate identity and α : G −→ 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 = C0(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 γs(y)(t) = αs(y(s −1 t)) for y ∈ D and s, t ∈ G. Recall that B is said to be * -regular if the natural mapping I ∈ Prim C * (B) → I ∩ B ∈ 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.Mathematics Subject Classification (2010). Primary 22D15, 22D20, 22D25, 43A20.