Given a normal subgroup bundle $\mathcal A$ of the isotropy bundle of a groupoid $Σ$, we obtain a twisted action of the quotient groupoid $Σ/\mathcal A$ on the bundle of group $C^*$-algebras determined by $\mathcal A$ whose twisted crossed product recovers the groupoid $C^*$-algebra $C^*(Σ)$. Restricting to the case where $\mathcal A$ is abelian, we describe $C^*(Σ)$ as the $C^*$-algebra associated to a $\mathbf T$-groupoid over the tranformation groupoid obtained from the canonical action of $Σ/\mathcal A$ on the Pontryagin dual space of $\mathcal A$. We give some illustrative examples of this result.
Given an action of a groupoid by isomorphisms on a Fell bundle (over another groupoid), we form a semidirect-product Fell bundle, and prove that its $C^{*}$-algebra is isomorphic to a crossed product.
We establish conditions under which the universal and reduced norms coincide for a Fell bundle over a groupoid. Specifically, we prove that the full and reduced C*-algebras of any Fell bundle over a measurewise amenable groupoid coincide, and also that for a groupoid G whose orbit space is T_0, the full and reduced algebras of a Fell bundle over G coincide if the full and reduced algebras of the restriction of the bundle to each isotropy group coincide.
Given a free and proper action of a groupoid on a Fell bundle (over another groupoid), we give an equivalence between the semidirect-product and the generalized-fixed-point Fell bundles, generalizing an earlier result where the action was by a group. As an application, we show that the Stabilization Theorem for Fell bundles over groupoids is essentially another form of crossed-product duality.
The author provides some definitions and structural results about Fell bundles, defined as C^*-algebra bundles over topological groupoids. Such bundles are a mutual generalization of semi-direct products of groups with C^*-algebras and C^*-algebra bundles over topological spaces. In particular a Morita equivalence theorem with semi-direct products is established.
We define the Zappa-Szép product of a Fell bundle by a groupoid, which turns out to be a Fell bundle over the Zappa-Szép product of the underlying groupoids. Under certain assumptions, every Fell bundle over the Zappa-Szép product of groupoids arises in this manner. We then study the representation associated with the Zappa-Szép product Fell bundle and show its relation to covariant representations. Finally, we study the associated universal C*-algebra, which turns out to be a C*-blend, generalizing an earlier result about the Zappa-Szép product of groupoid C*-algebras. In the case of discrete…
In this paper, we show that if E is a Fell bundle over an amenable étale locally compact Hausdorff groupoid such that every fiber on the unit space is nuclear, then $C?^*r(E)$ is also nuclear. In order to show this result, we introduce (minimal) tensor products of Fell bundles with fixed $C^*$-algebras.
We investigate some consequences of a recent stabilization result of Ionescu, Kumjian, Sims, and Williams, which says that every Fell bundle $C^*$-algebra is Morita equivalent to a canonical groupoid crossed product. First we use the theorem to give conditions that guarantee the $C^*$-algebras associated to a Fell bundle are either nuclear or exact. We then show that a groupoid is exact if and only if it is "Fell exact", in the sense that any invariant ideal gives rise to a short exact sequence of reduced Fell bundle $C^*$-algebras. As an application, we show that extensions of exact groupoids…
We consider fiberwise singly generated Fell-bundles over etale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo-Martin-Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the cross-sectional algebra in terms of measurable fields of traces on the C*-algebras of the restrictions of the Fell bundle to the isotropy subgroups of the groupoid. As a special case, we obtain a des…
We generalize Renault's notion of measurewise amenability to actions of second countable, Hausdorff, étale groupoids on separable $C^*$-algebras and show that measurewise amenability characterizes nuclearity of the crossed product whenever the $C^*$-algebra acted on is nuclear. In the more general context of Fell bundles over second countable, Hausdorff, étale groupoids, we introduce a version of Exel's approximation property. We prove that the approximation property implies nuclearity of the cross-sectional algebra whenever the unit bundle is nuclear. For Fell bundles associated to groupoid a…
Given a continuous open surjective morphism $π:G\to H$ of étale groupoids with amenable kernel, we construct a Fell bundle $E$ over $H$ and prove that its C*-algebra $C^*_r(E)$ is isomorphic to $C^*_r(G)$. This is related to results of Fell concerning C*-algebraic bundles over groups. The case $H=X$, a locally compact space, was treated earlier by Ramazan. We conclude that $C^*_r(G)$ is strongly Morita equivalent to a crossed product, the C*-algebra of a Fell bundle arising from an action of the groupoid $H$ on a C*-bundle over $H^0$. We apply the theory to groupoid morphisms obtained from ext…
Given a saturated Fell bundle A over an inverse semigroup S which is semi-abelian in the sense that the fibers over the idempotents of S are commutative, we construct a twisted etale groupoid L such that A can be recovered from L in a canonical way. As an application we recover most of Renault's recent result on the classification of Cartan subalgebras of C*-algebras through twisted etale groupoids.
We study the $C^*$-algebras associated to upper-semicontinuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer--Raeburn "Stabilization Trick," we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced $C^*$-algebras of any saturated upper-semicontinuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the $C^*$-algebra …
We define inverse semigroup actions on topological groupoids by partial equivalences. From such actions, we construct saturated Fell bundles over inverse semigroups and non-Hausdorff étale groupoids. We interpret these as actions on C*-algebras by Hilbert bimodules and describe the section algebras of these Fell bundles. Our constructions give saturated Fell bundles over non-Hausdorff étale groupoids that model actions on locally Hausdorff spaces. We show that these Fell bundles are usually not Morita equivalent to an action by automorphisms. That is, the Packer-Raeburn Stabilisation Trick doe…
We define the notion of a principal S-bundle where S is a groupoid group bundle and show that there is a one-to-one correspondence between principal S-bundles and elements of a sheaf cohomology group associated to S. We also define the notion of a locally unitary action and show that the spectrum of the crossed product is a principal bundle. Furthermore, we prove that the isomorphism class of the spectrum determines the exterior equivalence class of the action and that every principal bundle can be realized as the spectrum of some locally unitary crossed product.
We present a number of findings concerning groupoid dynamical systems and groupoid crossed products. The primary result is an identification of the spectrum of the groupoid crossed product when the groupoid has continuously varying abelian stabilizers and a well behaved orbit space. In this case, the spectrum of the crossed product is homeomorphic, via an induction map, to a quotient of the spectrum of the crossed product by the stabilizer group bundle. The main theorem is also generalized in the groupoid algebra case to an identification of the primitive ideal space. This generalization repla…
We analyse extensions $Σ$ of groupoids $G$ by bundles $A$ of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid $G$ by a given bundle $A$. There is a natural action of $Σ$ on the dual of $A$, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of $A$ with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of $G$ on the dual of $A$. We prove that the…
We show that every groupoid C*-algebra is isomorphic to its opposite, and deduce that there exist C*-algebras that are not stably isomorphic to groupoid C*-algebras, though many of them are stably isomorphic to twisted groupoid C*-algebras. We also prove that the opposite algebra of a section algebra of a Fell-bundle over a groupoid is isomorphic to the section algebra of a natural opposite bundle.
Consider a product system over the positive cone of a quasi-lattice ordered group. We construct a Fell bundle over an associated groupoid so that the cross-sectional algebra of the bundle is isomorphic to the Nica-Toeplitz algebra of the product system. Under the additional hypothesis that the left actions in the product system are implemented by injective homomorphisms, we show that the cross-sectional algebra of the restriction of the bundle to a natural boundary subgroupoid coincides with the Cuntz-Nica-Pimsner algebra of the product system. We apply these results to improve on existing suf…
We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.
This paper studies injective envelopes of groupoid dynamical systems and the corresponding boundaries. Analogue to the group case, we associate a bundle of compact Hausdorff spaces to any (discrete) groupoid (the Hamana boundary of the groupoid). We study bundles of compact topological spaces equipped with an action of a groupoid. We show that any groupoid has a minimal boundary (the Furstenberg boundary of the groupoid). We prove that the Hamana and Furstenberg boundaries are the same, for (discrete) groupoids. We find the relation between the reduced crossed product of the $\G$-injective env…
We show that if E is an equivalence of upper semicontinuous Fell bundles B and C over groupoids, then there is a linking bundle L(E) over the linking groupoid L such that the full cross-sectional algebra of L(E) contains those of B and C as complementary full corners, and likewise for reduced cross-sectional algebras. We show how our results generalise to groupoid crossed-products the fact, proved by Quigg and Spielberg, that Raeburn's symmetric imprimitivity theorem passes through the quotient map to reduced crossed products.
Given a Fell bundle $\mathscr C\overset{q}{\to}Ξ$ over the discrete groupoid $Ξ$, we study the symmetry of the associated Hahn algebra $\ell^{\infty,1}(Ξ\!\mid\!\mathscr C)$ in terms of the isotropy subgroups of $Ξ$. We prove that $Ξ$ is symmetric (resp. hypersymmetric) if and only if all of the isotropy subgroups are symmetric (resp. hypersymmetric). We also characterize hypersymmetry using Fell bundles with constant fibers, showing that for discrete groupoids, 'hypersymmetry' equals 'rigid symmetry'.
We examine a semigroup analogue of the Kumjian-Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we show how to represent semigroups with distinguished normal subsemigroups as `slice-sections' of groupoid bundles.
If $p \colon \mathcal B\to G$ is a Fell bundle over an étale groupoid, then we show that there is an norm reducing injective linear map $j \colon C^*_r(G;\mathcal B)\to Γ_{0}(G;\mathcal B)$ generalizing the well know map $j \colon C^*_{r}(G)\to C_{0}(G)$ in the case of an étale groupoid.
We show that Haefliger's cohomology for etale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Cech) cohomology for topological simplicial spaces.
In this paper, we show that the generalized fixed-point algebra of a proper groupoid dynamical system, under certain assumptions, may be fibered over any locally compact Hausdorff space to which a continuous map exists from the unit space of the underlying groupoid. We will also provide some important examples.
Coactions of Hopf C*-bimodules simultaneously generalize coactions of Hopf C*-algebras and actions of groupoids. Following an approach of Baaj and Skandalis, we construct reduced crossed products and establish a duality for fine coactions. Examples of coactions arise from Fell bundles on groupoids and actions of a groupoid on bundles of C*-algebras. Continuous Fell bundles on an etale groupoid correspond to coactions of the reduced groupoid algebra, and actions of a groupoid on a continuous bundle of C*-algebras correspond to coactions of the function algebra.
We define and study fibrations of topological groupoids. We interpret a groupoid fibration L->H with fibre G as an action of H on G by groupoid equivalences. Our main result shows that a crossed product for an action of L is isomorphic to an iterated crossed product first by G and then by H. Here groupoid action means a Fell bundle over the groupoid, and crossed product means the section C*-algebra.
We study the topology of the primitive ideal space of groupoid C*-algebras for groupoids with abelian isotropy. Our results include the known results for action groupoids with abelian stabilizers. Furthermore, we obtain complete results when the isotropy map is continuous except for jump discontinuities, and also when $G$ is a unit space fixing extension of a proper groupoid by an abelian group bundle. We hope that our methods will be a springboard to further results of this type.
We define a bundle over a totally disconnected set such that each fiber is homeomorphic to a fractal blowup. We prove that there is a natural action of a Renault-Deaconu groupoid on our fractafold bundle and that the resulting action groupoid is a Renault-Deaconu groupoid itself. We also show that when the bundle is locally compact the associated $C^*$-algebra is primitive and has a densely defined lower-semicontinuous trace.
Inspired by recent papers on twisted $K$-theory, we consider in this article the question of when a twist $\mathcal{R}$ over a locally compact Hausdorff groupoid $\mathcal{G}$ (with unit space a CW-complex) admits a twisted vector bundle, and we relate this question to the Brauer group of $\mathcal{G}$. We show that the twists which admit twisted vector bundles give rise to a subgroup of the Brauer group of $\mathcal{G}$. When $\mathcal{G}$ is an étale groupoid, we establish conditions (involving the classifying space $B\mathcal{G}$ of $\mathcal{G}$) which imply that a torsion twist $\mathcal{…
We introduce a new notion of twisted actions of inverse semigroups and show that they correspond bijectively to certain regular Fell bundles over inverse semigroups, yielding in this way a structure classification of such bundles. These include as special cases all the stable Fell bundles. Our definition of twisted actions properly generalizes a previous one introduced by Sieben and corresponds to Busby-Smith twisted actions in the group case. As an application we describe twisted etale groupoid C*-algebras in terms of crossed products by twisted actions of inverse semigroups and show that Sie…
Let $G$ be a second countable, locally compact groupoid with Haar system, and let $\mathcal{A}$ be a bundle of $C^{\ast}$-algebras defined over the unit space of $G$ on which $G$ acts continuously. We determine conditions under which the associated crossed product $C^{\ast}(G;\mathcal{A})$ is a continuous trace $C^{\ast}$-algebra.
This article extends the main results of the publication arXiv:2001.01312 to the case of a twisted groupoid. More precisely, it gives a decomposition of the C*-algebra of a twisted locally compact groupoid with Haar system in presence of a normal subgroupoid. When the normal subgroupoid and the twist over it are abelian, one obtains another twisted groupoid C*-algebra. This is applied to C*-algebraic deformation quantization and to multiplier representations of locally compact abelian groups.
Let G be a Lie groupoid and L his Lie algebroid. We give a definition of the classical limit of a C^*-bundle and we use the tangent groupoid associated to G to show that the Poisson structure on L is the classical limit of a C^*-bundle.
We construct a locally compact groupoid with the properties in the title. Our example is based closely on constructions used by Higson, Lafforgue, and Skandalis in their work on counterexamples to the Baum-Connes conjecture. It is a bundle of countable groups over the one point compactification of the natural numbers, and is Hausdorff, second countable and étale.
The second author showed how Katsura's construction of the C*-algebra of a topological graph E may be twisted by a Hermitian line bundle L over the edge space E. The correspondence defining the algebra is obtained as the completion of the compactly supported continuous sections of L. We prove that the resulting C*-algebra is isomorphic to a twisted groupoid C*-algebra where the underlying groupoid is the Renault-Deaconu groupoid of the topological graph with Yeend's boundary path space as its unit space.
We develop a framework suitable for obtaining simplicity criteria for reduced $C^*$-algebras of Hausdorff etale groupoids. This is based on the study of certain non-degenerate $C^*$-subalgebras (in the case of groupoids, the $C^*$-algebra of the interior isotropy bundle), for which one can control (non-unique) state extensions to the ambient C*-algebra. As an application, we give simplicity criteria for reduced crossed products of abelian $C^*$-algebras by discrete groups.
We study the C*-algebras of Fell bundles. In particular, we prove the analogue of Renault's disintegration theorem for groupoids. As in the groupoid case, this result is the key step in proving a deep equivalence theorem for the C*-algebras of Fell bundles.
Topological T-duality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dual-torus bundle. We give a new geometric construction of T-dualization, which allows the duality to be extended in following two directions. First, bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized. Second, bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated, though in this case the base space of the bundles is best viewed as a topological stack. Some methods developed for the con…
The adiabatic groupoid $\mathcal{G}_{ad}$ of a smooth groupoid $\mathcal{G}$ is a deformation relating $\mathcal{G}$ with its algebroid. In a previous work, we constructed a natural action of $\mathbb{R}$ on the C*-algebra of zero order pseudodifferential operators on $\mathcal{G}$ and identified the crossed product with a natural ideal $J(\mathcal{G})$ of $C^*(\mathcal{G}_{ad})$. In the present paper we show that $C^*(\mathcal{G}_{ad})$ itself is a pseudodifferential extension of this crossed product in a sense introduced by Saad Baaj. Let us point out that we prove our results in a slightly …
Given a locally compact abelian group $G$, we give an explicit formula for the Dixmier--Douady invariant of the $C^*$-algebra of the groupoid extension associated to a Čech $2$-cocycle in the sheaf of germs of continuous $G$-valued functions. We then exploit the blow-up construction for groupoids to extend this to some more general central extensions of étale equivalence relations.
We introduce the notion of groupoid pre-equivalences and prove that they give rise to groupoid equivalences by taking certain quotients. Then, given an equivalence of Fell bundles $\mathscr{B}$ and $\mathscr{C}$ and another equivalence between $\mathscr{C}$ and $\mathscr{D}$, we construct an equivalence between $\mathscr{B}$ and $\mathscr{D}$ out of the tensor product bundle. As a consequence, we obtain that Fell bundle equivalence is indeed an equivalence relation.
We study Fell bundles on groupoids from the viewpoint of quantale theory. Given any saturated upper semicontinuous Fell bundle $π:E\to G$ on an étale groupoid $G$ with $G_0$ locally compact Hausdorff, equipped with a suitable completion C*-algebra $A$ of its convolution algebra, we obtain a map of involutive quantales $p:\mathrm{Max}\ A\toΩ(G)$, where $\mathrm{Max}\ A$ consists of the closed linear subspaces of $A$ and $Ω(G)$ is the topology of $G$. We study various properties of $p$ which mimick, to various degrees, those of open maps of topological spaces. These are closely related to proper…
