Multiply generated dynamical systems and the duality of higher rank graph algebras

We define a semidirect product groupoid of a system of partially defined local homeomorphisms T = (T1, . . . , Tr). We prove that this construction gives rise to amenable groupoids. The associated algebra is a Cuntz-like algebra. We use this construction for higher rank graph algebras in order to give a topological interpretation for the duality in E-theory between C∗(Λ) and C∗(Λop). Introduction Toeplitz algebras have been used to define extensions of C-algebras. The beginning of this paper is in [19] where a Toeplitz algebra was the main tool in constructing a K-homology class for higher rank graph algebras. Let (Λ, σ) be a higher rank graph with shape σ (see [16]), Λ the set of morphism of nonzero shape and Λ = {Ω} ∪Λ where Ω is a symbol (the vacuum morphism) which does not belong to Λ. We define left and right creations on the Fock space FΛ = F = l (Λ): Lλδμ = { δλμ if s(λ) = t(μ) 0 otherwise, Rλδμ = { δμλ if s(μ) = t(λ) 0 otherwise. LλΩ = RλΩ = δλ Supportted by the grant 2-CEx06-11-34/2006