*-regularity of certain generalized group algebras |
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Authors: | Chi-Wai Leung |
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Institution: | 1.Department of Mathematics,The Chinese University of Hong Kong,Shatin,Hong Kong |
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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. |
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