The paper establishes, for a wide class of locally compact groupoids $Γ$, the E-theoretic descent functor at the $C^{*}$-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. The second section shows that $Γ$-actions on a $C_{0}(X)$-algebra $B$, where $X$ is the unit space of $Γ$, can be usefully formulated in terms of an action on the associated bundle $B^{\sharp}$. The third section shows that the functor $B\to C^{*}(Γ,B)$ is continuous and exact, and uses the disintegration theory of J. Renault. The last section establishes the existence of the descent functor under a very mild condition on $Γ$, the main technical difficulty involved being that of finding a $Γ$-algebra that plays the role of C_{b}(T,B)^{cont}$ in the group case.
We study the connection between the Baum-Connes conjecture for an ample groupoid $G$ with coefficient $A$ and the Künneth formula for the K-theory of tensor products by the crossed product $A\rtimes_r G$. To do so we develop the machinery of Going-Down functors for ample groupoids. As an application we prove that both the uniform Roe algebra of a coarse space which uniformly embeds into a Hilbert space and the maximal Roe algebra of a space admitting a fibred coarse embedding into a Hilbert space satisfy the Künneth formula. We also provide a stability result for the Künneth formula using cont…
We prove that relative functors out of a cofibration category are essentially the same as relative functors which are only defined on the subcategory of cofibrations. As an application we give a new construction of the functor that assigns to a groupoid its groupoid $C^*$-algebra and thereby its topological $K$-theory spectrum.
We construct a large class of morphisms, which we call partial morphisms, of groupoids that induce $*$-morphisms of maximal and minimal groupoid $C^*$-algebras. We show that the association of a groupoid to its maximal (minimal) groupoid $C^*$-algebra and the association of a partial morphism to its induced morphism are functors (both of which extend the Gelfand functor). We show how to geometrically visualize lots of $*$-morphisms between groupoid $C^*$-algebras. As an application, we construct a groupoid models of the entire inductive systems of the Jiang-Su algebra $\mathcal{Z}$ and the Raz…
We study a Going-Down (or restriction) principle for ample groupoids and its applications. The Going-Down principle for locally compact groups was developed by Chabert, Echterhoff and Oyono-Oyono and allows to study certain functors, that arise in the context of the topological K-theory of a locally compact group, in terms of their restrictions to compact subgroups. We extend this principle to the class of ample Hausdorff groupoids using Le Gall's groupoid equivariant version of Kasparov's bivariant KK-theory. Moreover, we provide an application to the Baum-Connes conjecture for ample groupoid…
We classify which dual functors on a unitary multitensor category are compatible with the dagger structure in terms of groupoid homomorphisms from the universal grading groupoid to $\mathbb{R}_{>0}$ where the latter is considered as a groupoid with one object. We then prove that all unitary dual functors induce unitarily equivalent bi-involutive structures. As an application, we provide the unitary version of the folklore correspondence between shaded planar ${\rm C^*}$ algebras with finite dimensional box spaces and unitary multitensor categories with a chosen unitary dual functor and chosen …
Given a higher-rank graph $Λ$, we investigate the relationship between the cohomology of $Λ$ and the cohomology of the associated groupoid $G_Λ$. We define an exact functor between the abelian category of right modules over a higher-rank graph $Λ$ and the category of $G_Λ$-sheaves, where $G_Λ$ is the path groupoid of $Λ$. We use this functor to construct compatible homomorphisms from both the cohomology of $Λ$ with coefficients in a right $Λ$-module, and the continuous cocycle cohomology of $G_Λ$ with values in the corresponding $G_Λ$-sheaf, into the sheaf cohomology of $G_Λ$.
In this paper we define K-theoretic secondary invariants attached to a Lie groupoid $G$. The K-theory of $C^*_r(G_{ad}^0)$ (where $G_{ad}^0$ is the adiabatic deformation $G$ restricted to the interval $[0,1)$) is the receptacle for K-theoretic secondary invariants. We give a Lie groupoid version of construction given by Piazza and Schick in the setting of the Coarse Geometry. Our construction directly generalises to more involved geometrical situation, such as foliations, well encoded by a Lie groupoid. Along the way we tackle the problem of producing a wrong-way functoriality between adiabati…
A locally compact groupoid is said to be exact if its associated reduced crossed product functor is exact. In this paper, we establish some permanence properties of exactness, including generalizations of some known results for exact groups. Our primary goal is to show that exactness descends to certain types of closed subgroupoids, which in turn gives conditions under which the isotropy groups of an exact groupoid are guaranteed to be exact. As an initial step toward these results, we establish the exactness of any transformation groupoid associated to an action of an exact groupoid on a loca…
We study strongly graded groupoids, which are topological groupoids $\mathcal G$ equipped with a continuous, surjective functor $κ: \mathcal G \to Γ$, to a discrete group $Γ$, such that $κ^{-1}(γ)κ^{-1}(δ) = κ^{-1}(γδ)$, for all $γ, δ\in Γ$. We introduce the category of graded $\mathcal G$-sheaves, and prove an analogue of Dade's Theorem: $\mathcal G$ is strongly graded if and only if every graded $\mathcal G$-sheaf is induced by a $\mathcal G_ε$-sheaf. The Steinberg algebra of a graded ample groupoid is graded, and we prove that the algebra is strongly graded if and only if the groupoid is. A…
