Groupoid dynamical systems and crossed product

Abstract

By analogy with W*-dynamical system, we define a W*-groupoid dynamical system (M, Γ, ρ) where M is a W*-algebra, Γ is a locally compact measured groupoid, and ρ:Γ → Aut(M) is a continuous groupoid homorphism. The groupoid crossed product M×_ρΓ is defined by making use of the non-commutative integration theory of A. Connes i.e. integration theory over singular quotient spaces, 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 (L^∞(G), Γ, ρ). In this case, there exists a co-action $\hat \rho$ of G on End_Λ(Γ) and the groupoid crossed product L^∞(G)×_ρΓ is isomorphic to the co-crossed product $End_\Lambda (\Gamma ) * _{\hat \rho } G$ of End_Λ(Γ) by G in the sense of Nakagami and Takesaki. Similar results hold for the C*-algebraic framework. This note is a short report on [7], [8].