This paper contains a quite detailed description of the C*-algebra arising from the transformation groupoid of a rational map of degree at least two on the Riemann sphere. The algebra is decomposed stepwise via extensions of familiar C*-algebras whose nature depend on the structure of the Julia set and the stable regions in the Fatou set, as well as on the behaviour of the critical points.
In this paper, we introduce properties including groupoid comparison, pure infiniteness and paradoxical comparison as well as a new algebraic tool called groupoid semigroup for locally compact Hausdorff étale groupoids. We show these new tools help establishing pure infiniteness of reduced groupoid $C^*$-algebras. As an application, we show a dichotomy of stably finiteness against pure infiniteness for reduced groupoid $C^*$-algebras arising from locally compact Hausdorff étale minimal topological principal groupoids. This generalizes the dichotomy obtained by Bönicke-Li and Rainone-Sims. We a…
Given a not-necessarily Hausdorff, topologically free, twisted étale groupoid $(G, L)$, we consider its "essential groupoid C*-algebra", denoted $C^*_{ess}(G, L)$, obtained by completing $C_c(G, L)$ with the smallest among all C*-seminorms coinciding with the uniform norm on $C_c(G^0)$. The inclusion of C*-algebras $(C_0(G^0), C^*_{ess}(G, L))$ is then proven to satisfy a list of properties characterizing it as what we call a "weak Cartan inclusion". We then prove that every weak Cartan inclusion $(A, B)$, with $B$ separable, is modeled by a topologically free, twisted étale groupoid, as above…
In this paper, we introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple proof that effectiveness of an étale groupoid is implied by the full uniqueness property of its groupoid C*-algebra.
We define what it means for a proper continuous morphism between groupoids to be Haar system preserving, and show that such a morphism induces (via pullback) a *-morphism between the corresponding convolution algebras. We proceed to provide a plethora of examples of Haar system preserving morphisms and discuss connections to noncommutative CW-complexes and interval algebras. We prove that an inverse system of groupoids with Haar system preserving bonding maps has a limit, and that we get a corresponding direct system of groupoid $C^*$-algebras. An explicit construction of an inverse system of …
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.
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.
Let G be second countable locally compact Hausdorff groupoid with a continuous Haar system. We remove the assumption of amenability in a theorem of Clark about groupoids whose $C^*$-algebras are CCR. We show that if the groupoid C*-algebra of G is CCR then the orbits of G are closed.
We describe a construction by Gábor Elek, associating C*-algebras with uniformly recurrent subgroups, in the language of groupoid C*-algebras. This allows us to simplify several proofs in the original paper and fully characterise their nuclearity. We furthermore relate our groupoids to the dynamics of the group acting on its uniformly recurrent subgroup.
We describe representations of groupoid C*-algebras on Hilbert modules over arbitrary C*-algebras by a universal property. For Hilbert space representations, our universal property is equivalent to Renault's Integration-Disintegration Theorem. For a locally compact group, it is related to the automatic continuity of measurable group representations. It implies known descriptions of groupoid C*-algebras as crossed products for étale groupoids and transformation groupoids of group actions on spaces.
We define a groupoid from a labelled space and show that it is isomorphic to the tight groupoid arising from an inverse semigroup associated with the labelled space. We then define a local homeomorphism on the tight spectrum that is a generalization of the shift map for graphs, and show that the defined groupoid is isomorphic to the Renault-Deaconu groupoid for this local homeomorphism. Finally, we show that the C*-algebra of this groupoid is isomorphic to the C*-algebra of the labelled space as introduced by Bates and Pask.
In this note we show that every Kirchberg algebra in the UCT class is the $C^{\ast}$-algebra of a Hausdorff, ample, amenable and locally contracting groupoid. The non-unital case follows from Spielberg's graph-based models for Kirchberg algebras. Our contribution is the unital case and our proof builds on Spielberg's construction.
We construct ample groupoids from certain categories of paths, and prove that their $C^*$-algebras coincide with the continued fraction AF algebras of Effros and Shen. The proof relies on recent classification results for simple nuclear $C^*$-algebras. The groupoids are not principal. This provides examples of Cartan subalgebras in the continued fraction AF algebras that are isomorphic, but not conjugate, to the standard diagonal subalgebras.
We compute the homology of the groupoid associated to the Katsura algebras, and show that they capture the $K$-theory of the $C^*$-algebras, and hence satisfying the (HK) conjecture posted by Matui. Moreover, we show that several classifiable simple $C^*$-algebras are groupoid $C^*$-algebras of this class.
We give a concise introduction to (discrete) algebras arising from étale groupoids, (aka Steinberg algebras) and describe their close relationship with groupoid C*-algebras. Their connection to partial group rings via inverse semigroups also explored.
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…
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.
Given a self-similar groupoid action $(G,E)$ on the path space of a finite graph, we study the associated Exel-Pardo étale groupoid ${\mathcal G}(G,E)$ and its $C^*$-algebra $C^*(G,E)$. We review some facts about groupoid actions, skew products and semi-direct products and generalize a result of Renault about similarity of groupoids which resembles Takai duality. We also describe a general strategy to compute the $K$-theory of $C^*(G,E)$ and the homology of ${\mathcal G}(G,E)$ in certain cases and illustrate with an example.
Let $C^*$-algebra that is acted upon by a compact abelian group. We show that if the fixed-point algebra of the action contains a Cartan subalgebra $D$ satisfying an appropriate regularity condition, then $A$ is the reduced $C^*$-algebra of a groupoid twist. We further show that the embedding $D \hookrightarrow A$ is uniquely determined by the twist. These results generalize Renault's results on Cartan subalgebras of $C^*$-algebras.
Suppose $\mathcal{G}$ is a second-countable locally compact Hausdorff étale groupoid, $G$ is a discrete group containing a unital subsemigroup $P$, and $c:\mathcal{G}\rightarrow G$ is a continuous cocycle. We derive conditions on the cocycle such that the reduced groupoid $C^*$-algebra $C_r^*(\mathcal{G})$ may be realised naturally as the covariance algebra of a product system over $P$ with coefficient algebra $C_r^*(c^{-1}(e))$. When $(G,P)$ is a quasi-lattice ordered group, we also derive conditions that allow $C_r^*(\mathcal{G})$ to be realised as the Cuntz--Nica--Pimsner algebra of a compa…
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 thi…
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.
