Banach Algebra Crossed Products And Their Representations

This dissertation is devoted to the investigation of Banach algebra crossed prod-ucts, including the universal property of Banach algebra crossed products and it’s ap-plications, reduced crossed products of Banach algebra, and locally m-convex algebra crossed products and their representations. It consists of four chapters.In the first chapter, we introduce the background, review the developments and achievements until now, and give preliminaries.In Chapter 2, we prove that Banach algebra crossed products have the required universal property. The main results of this chapter are as follows.Theorem A. Let (A, G, α) be a Banach algebra dynamical system, where A has a bounded approximate left identity, and let R be a non-empty uniformly bounded class of non-degenerate continuous covariant representations of (A, G, α) on Banach space X. Suppose that B is a Banach algebra such that(i) there is a covariant homomorphism (kA,kG) of (A,G,α) into Ml(B),(ii) given a non-degenerate R-continuous covariant representation (Ï€, U) of (A,G,α), there is a non-degenerate representation L= L(π，U) of B such that L o kA=Ï€ and L(？(kG= U,(iii) λ(B)= span{kA(a)kG(f):a∈A,f∈Cc(G)}, Then there is an isomorphism K:Ml(B)â†'(A×αG)R such thatTheorem B. Let (A,G,α) and (B,G,α) be two equivariantly isomorphic Banach algebra dynamical systems, φ:Aâ†'B be an equivariant isomorphism. Suppose that R1 is a non-empty uniformly bounded class of non-degenerate continuous covariant representations of (A,G,α), and R2 is a non-empty uniformly bounded class of non-degenerate continuous covariant representations of (B, G,β) such that Then (A ×α G)R1 is isomorphic to (B ×βG)R2.In Chapter 3, we define the reduced crossed products of Banach algebra dynamical systems. Under some conditions, we prove that the crossed products of a Banach algebra dvnamical svstem and its reduced crossed products are egual. The main results of this chapter are as follows.Theorem C. Let (A, G, α) be a bounded Banach algebra dynamical system. Let (Ï€,U)} be a continuous covariant representations of (A, G, α) on the Banach space X and suppose Let (Ï€, A) be the regular continuous covariant representations of (A, G, α) associated to Ï€ on L1(G,X). If G is amenable, then holds for all f∈Cc(G, A).In Chapter 4, we define the notion of locally m-convex algebra crossed product, and prove that every completely locally m-convex algebra crossed product is an inverse limit of the inverse system of Banach algebra crossed products by the inverse limit theory. Based on it, we study the representations of locally m-convex algebra dynam-ical systems and locally m-convex algebra crossed products. The main results of this chapter are as follows.Theorem D. Let (A, G, α) be a completely inverse locally m-convex algebra dynam-ical system, and R be a non-empty semi-uniformly bounded class of non-degenerate continuous covariant representations of (A,G,α). Then up to an algebraic and topological isomorphism.Theorem E. Let (A, G, α) be an inverse locally m-convex algebra dynamical system, where A has a bounded approximate left identity, and let R be a semi-uniformly bounded class of non-degenerate continuous covariant representations of (A, G, α), and let T be a non-degenerate continuous representation of (A ×α G)R on Banach space X. Then (T (?) iRA,T (?) iRG) is a non-degenerate continuous covariant representation of (A, G, α). That is, T (?) iRA is continuous, T (?) iRG is strongly continuous, and for all a E A, r ∈ G, where T is a continuous extension of T on Ml ((A ×α G)R).Theorem F. Let (A, G, α) be an inverse locally m-convex algebra dynamical system, where A has a bounded approximate left identity.(1) If (Ï€,U) is a non-degenerate continuous covariant representation of (A,G,α) on Banach space X, and ∞, then there is a non-degenerate continuous representation Ï€(?) U of L1(G, A, α) on X, such that for all f∈Cc(G, A, α).(2) If T is a non-degenerate continuous representation of L1(G, A, α) on Banach space X, then there is a continuous covariant representation of (A, G, α) on X, such that...