Groupoid Dynamical Systems and Crossed Product, II—The Case of C^*-Systems

抄録

By analogy with C^*-dynamical system, we define a C^*-groupoid dynamical system (A, Γ, ρ) where A is a C^*-algebra, Γ is a locally compact groupoid, and ρ: Γ→Aut(A) is a continuous groupoid homomorphism. The groupoid crossed product A×_ρΓ is defined and is shown to have similar properties as the case of a group action. As a special case of this situation, if ρ is a continuous homomorphism from Γ to a locally compact group G, we obtain groupoid dynamical system (C₀(G), Γ, ρ). In this case, there exists a co-action \hat{ρ} of G on C^*(Γ) and the groupoid crossed product C₀(G)×_ρΓ is isomorphic to the co-crossed product C^*(Γ)*_\hat{ρ}G of C^*(Γ) by G. The results in this paper is obtained by the analogy with our previous results for the case of W^*-systems.