Let B be a *-semisimple Banach algebra with a bounded approximate identity and \documentclass[12pt]{minimal}
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\begin{document}$${\alpha: G \longrightarrow {\rm Aut}_{*}(B)}$$\end{document} (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−1t)) for \documentclass[12pt]{minimal}
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\begin{document}$${y \in D}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${s, t \in G}$$\end{document} . Recall that B is said to be *-regular if the natural mapping \documentclass[12pt]{minimal}
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\begin{document}$${I\in {\rm Prim} \, C^{*}(B) \mapsto I\cap B\in {\rm Prim}_{*}(B)}$$\end{document} is a homeomorphism under the hull-kernel topology. When G is amenable, we show that if B is *-regular, then the generalized group algebra L1(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.
