We describe the envelope C*-algebra associated to a partial action of a countable discrete group on a locally compact space as a groupoid C*-algebra (more precisely as a C*-algebra from an equivalence relation) and we use our approach to show that, for a large class of partial actions of Z on the Cantor set, the envelope C*-algebra is an AF-algebra. We also completely characterize partial actions of a countable discrete group on a compact space such that the envelope action acts in a Hausdorff space.
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 consider the amenability of groupoids $G$ equipped with a group valued cocycle $c:G\to Q$ with amenable kernel $c^{-1}(e)$. We prove a general result which implies, in particular, that $G$ is amenable whenever $Q$ is amenable and if there is countable set $D\subset G$ such that $c(G^{u})D=Q$ for all $u\in G^{(0)}$. We show that our result is applicable to groupoids arising from partial semigroup actions. We explore these actions in detail and show that these groupoids include those arising from directed graphs, higher rank graphs and even topological higher rank graphs. We believe our metho…
In this paper, we investigate primeness of groupoid graded rings. We provide a set of necessary and sufficient conditions for primeness of a nearly-epsilon strongly groupoid graded ring. Furthermore, we apply our main result to get a characterization of prime partial skew groupoid rings, and in particular of prime groupoid rings, thereby generalizing a classical result by Connell and partially generalizing recent results by Steinberg.
We give a new very concrete description of the C*-envelope of the tensor algebra associated to multivariable dynamical system. In the surjective case, this C*-envelope is described as a crossed product by an endomorphism, and as a groupoid C*-algebra. In the non-surjective case, it is a full corner of a such an algebra. We also show that when the space is compact, then the C*-envelope is simple if and only if the system is minimal.
In this paper, we provide two types of boundary path groupoids from a generalized Boolean dynamical system $(\mathcal{B},\mathcal{L}, θ, \mathcal{I}_α)$. For the first groupoid, we associate an inverse semigroup to a generalized Boolean dynamical system and use the tight spectrum $\mathsf{T}$ as the unit space of a groupoid $Γ(\mathcal{B},\mathcal{L}, θ, \mathcal{I}_α)$ that is isomorphic to the tight groupoid $\mathcal{G}_{tight}$. The other one is defined as the Renault-Deaconu groupoid $Γ(\partial E, σ_E)$ arising from a topological correspondence $E$ associated with a generalized Boolean d…
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 groupoid actions on left cancellative small categories and their associated Zappa-Szép products. We show that certain left cancellative small categories with nice length functions can be seen as Zappa-Szép products. We compute the associated tight groupoids, characterizing important properties of them, like being Hausdorff, effective and minimal. Finally, we determine amenability of the tight groupoid under mild, reasonable hypotheses.
Given a partial action $α$ of a connected groupoid $\mathcal{G}$ on an associative ring $A$ we investigate under what conditions the partial skew groupoid ring $A\star_α\mathcal{G}$ can be realized as a partial skew group ring. In such a case applications concerning to the separability, semisimplicity and Frobenius property of the ring extension $A\subset A\star_α\mathcal{G}$ as well as to the artinianity of $A\star_α\mathcal{G}$ are given.
We consider an inclusion $B\subseteq M$ of finite von Neumann algebras satisfying $B'\cap M\subseteq B$. A partial isometry $v\in M$ is called a groupoid normalizer if $vBv^*, v^*Bv\subseteq B$. Given two such inclusions $B_i\subseteq M_i$, $i=1,2$, we find approximations to the groupoid normalizers of $B_1 \vnotimes B_2$ in $M_1\vnotimes M_2$, from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail…
We obtain a characterization in terms of dynamical systems of those r-discrete groupoids for which the groupoid C*-algebra is approximately finite-dimensional (AF). These ideas are then used to compute the K-theory for AF algebras by utilizing the actions of these partial homeomorphisms, and these K-theoretic calculations are applied to some specific examples of AF algebras. Finally, we show that, for a certain class of dimension groups, a groupoid can be obtained directly from the dimension group's structure whose associated C*-algebra has K_0 group isomorphic to the original dimension group.
We introduce the notion of continuous orbit equivalence for partial dynamical systems, and give an equivalent characterization in terms of Cartan-isomorphisms for partial C*-crossed products. Both graph C*-algebras and semigroup C*-algebras can be described as C*-algebras attached to partial dynamical systems. As applications, for graphs, we generalize and explain a result of Matsumoto and Matui relating orbit equivalence and Cartan-isomorphism, and for semigroups, we strengthen several structural results for semigroup C*-algebras concerning amenability, nuclearity as well as simplicity of bou…
In this paper we show that the universal C*-algebra satisfying the Cuntz-Li relations is generated by an inverse semigroup of partial isometries. We apply Exel's theory of tight representations to this inverse semigroup. We identify the universal C*-algebra as the C*-algebra of the tight groupoid associated to the inverse semigroup.
Given a locally convex space $(\mathcal{A},τ)$ with a Hausdorff locally convex topology $τ$ such that the following maps are continuous; $u \mapsto u^*$ for all $u \in \mathcal{A}$, $x \mapsto x\cdot y$ and $x \mapsto z\cdot x$ for every left and right multipliers of $\mathcal{A}$. In this paper we re-characterized the locally convex partial $*$-algebra $(\mathcal{A}, Γ,\cdot,*,τ)$ arising from these continuous maps in terms of convolution algebra of a Lie groupoid $Γ\rightrightarrows \mathcal{A}$. This is advantageous because the pathologies of the underlying spaces owing to their quantum mec…
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…
Using the calculus of Fourier integral operators on Lie groupoids developped in [18], we study the fundamental solution of the evolution equation ($\partial$ $\partial$t + iP)u = 0 where P is a self adjoint elliptic order one G-pseudodifferential operator on the Lie groupoid G. Along the way, we continue the study of distributions on Lie groupoids done in [17] by adding the reduced C *-algebra of G in the picture and we investigate the local nature of the regularizing operators of [32].
Let $α= \{ α_g : R_{g^{-1}} \rightarrow R_g \}_{g \in \textrm{mor}(G)}$ be a partial action of a groupoid $G$ on a non-associative ring $R$ and let $S = R \star_α G$ be the associated partial skew groupoid ring. We show that if $α$ is global and unital, then $S$ is left (right) artinian if and only if $R$ is left (right) artinian and $R_g = \{ 0 \},$ for all but finitely many $g \in \textrm{mor}(G)$. We use this result to prove that if $α$ is unital and $R$ is alternative, then $S$ is left (right) artinian if and only if $R$ is left (right) artinian and $R_g = \{ 0 \},$ for all but finitely ma…
In this work we realize Leavitt path algebras as partial skew groupoid rings. This yields a free path groupoid grading on Leavitt path algebras. Using this grading we characterize free path groupoid graded isomorphisms of Leavitt path algebras that preserves generators.
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.
Given a partial action $α$ of a groupoid $G$ on a ring $R$, we study the associated partial skew groupoid ring $R \rtimes_α G$, which carries a natural $G$-grading. We show that there is a one-to-one correspondence between the $G$-invariant ideals of $R$ and the graded ideals of the $G$-graded ring $R \rtimes_αG.$ We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of $R \rtimes_αG.$ We show that every ideal of $R \rtimes_αG$ is graded if, and only if, $α$ has the residual intersection property. Furthermore, if $α$ is induced by a topological …
Given an inverse semigroup $S$ endowed with a partial action on a topological space $X$, we construct a groupoid of germs $S\ltimes X$ in a manner similar to Exel's groupoid of germs, and similarly a partial action of $S$ on an algebra $A$ induces a crossed product $A\rtimes S$. We then prove, in the setting of partial actions, that if $X$ is locally compact Hausdorff and zero-dimensional, then the Steinberg algebra of the groupoid of germs $S\ltimes X$ is isomorphic to the crossed product $A_R(X)\rtimes S$, where $A_R(X)$ is the Steinberg algebra of $X$. We also prove that the converse holds,…
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.
Using an algebraic point of view we present an introduction to the groupoid theory, that is, we give fundamental properties of groupoids as, uniqueness of inverses and properties of the identities, and study subgroupoids, wide subgroupoids and normal subgroupoids. We also present the isomorphism theorems for groupoids and as an applications we obtain the corresponding version of Zassenhaus Lemma and Jordan-Hölder Theorem for groupoids. Finally inspired by the Ehresmann-Schein-Nambooripad Theorem we improve a result of R. Exel concerning a one to one correspondence between partial actions of gr…
We study the interplay between Steinberg algebras and partial skew rings: For a partial action of a group in a Hausdorff, locally compact, totally disconnected topological space, we realize the associated partial skew group ring as a Steinberg algebra (over the transformation groupoid attached to the partial action). We then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings, over commutative algebras, in terms of continuous orbit equivalence of the associated partial actions. Finally, we show that any Steinberg algebra, associated to a Hausdorf…
We consider a version of the notion of F-inverse semigroup (studied in the algebraic theory of inverse semigroups). We point out that an action of such an inverse semigroup on a locally compact space has associated a natural groupoid construction, very similar to the one of a transformation group. We discuss examples related to Toeplitz algebras on subsemigroups of discrete groups,to Cuntz-Krieger algebras, and to crossed-products by partial automorphisms in the sense of Exel.
An ultragraph gives rise to a labelled graph with some particular properties. In this paper we describe the algebras associated to such labelled graphs as groupoid algebras. More precisely, we show that the known groupoid algebra realization of ultragraph C*-algebras is only valid for ultragraphs for which the range of each edge is finite, and we extend this realization to any ultragraph (including ultragraphs with sinks). Using our machinery, we characterize the shift space associated to an ultragraph as the tight spectrum of the inverse semigroup associated to the ultragraph (viewed as a lab…
The Cuntz algebra O_n is presented as a partial crossed product in which an amenable group partially acts on an abelian C*-algebra. The partial action is related to the Cuntz groupoid for O_n and connections are made with non-self-adjoint subalgebras of O_n, particularly the Volterra nest subalgebra. These ideas are also extended to the M_k(O_n) context.
We prove a spectral theorem for bimodules in the context of graph C*-algebras. A bimodule over a suitable abelian algebra is determined by its spectrum (i.e., its groupoid partial order) iff it is generated by the Cuntz-Krieger partial isometries which it contains iff it is invariant under the gauge automorphisms. We study 1-cocycles on the Cuntz-Krieger groupoid associated with a graph C*-algebra, obtaining results on when integer valued or bounded cocycles on the natural AF subgroupoid extend. To a finite graph with a total order, we associate a nest subalgebra of the graph C*-algebra and th…
Partial actions of discrete abelian groups can be used to construct both groupoid C*-algebras and partial crossed product algebras. In each case there is a natural notion of an analytic subalgebra. We show that for countable subgroups of the real numbers and free partial actions, these constructions yield the same C*-algebras and the same analytic subalgebras. We also show that under suitable hypotheses an analytic partial crossed product preserves all the information in the dynamical system in the sense that two analytic partial crossed products are isomorphic as Banach algebras if, and only …
We extend ultragraph shift spaces and the realization of ultragraph C*-algebras as partial crossed products to include ultragraphs with sinks (under a mild condition, called (RFUM2), which allow us to dismiss the use of filters) and we describe the associated transformation groupoid. Using these characterizations we study continuous orbit equivalence of ultragraph shift spaces (via groupoids) and KMS and ground states (via partial crossed products).
The groupoid normalisers of a unital inclusion $B\subseteq M$ of von Neumann algebras consist of the set $\mathcal{GN}_M(B)$ of partial isometries $v\in M$ with $vBv^*\subseteq B$ and $v^*Bv\subseteq B$. Given two unital inclusions $B_i\subseteq M_i$ of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion $B_1\ \overline{\otimes}\ B_2\subseteq M_1\ \overline{\otimes}\ M_2$ establishing the formula $$ \mathcal{GN}_{M_1\,\overline{\otimes}\,M_2}(B_1\ \overline{\otimes}\ B_2)''=\mathcal{GN}_{M_1}(B_1)''\ \overline{\otimes}\ \mathcal{GN}_{M_2}(B_2)'' $$ when one i…
We define a semidirect product groupoid of a system of partially defined local homeomorphisms $T=(T_{1},..., T_{r})$. 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 $C^{*}(Λ^{op})$.
In this paper, we apply the theory of inverse semigroups to the $C^{*}$-algebra $U[\mathbb{Z}]$ considered in \cite{Cuntz}. We show that the $C^{*}$-algebra $U[\mathbb{Z}]$ is generated by an inverse semigroup of partial isometries. We explicity identify the groupoid $\mathcal{G}_{tight}$ associated to the inverse semigroup and show that $\mathcal{G}_{tight}$ is exactly the same groupoid obtained in \cite{Cuntz-Li}.
For a free partial action of a group in a set we realize the associated partial skew group ring as an algebra of functions with finite support over an equivalence relation and we use this result to characterize the ideals in the partial skew group ring. This generalizes, to the purely algebraic setting, the known characterization of partial C*-crossed products as groupoid C*-algebras. For completeness we include a new proof of the C* result for free partial actions.
We study the unitary orbit of a normal operator $a\in \mathcal B(\mathcal H)$, regarded as a homogeneous space for the action of unitary groups associated with symmetrically normed ideals of compact operators. We show with an unified treatment that the orbit is a submanifold of the differing ambient spaces if and only if the spectrum of $a$ is finite, and in that case it is a closed submanifold. For arithmetically mean closed ideals, we show that nevertheless the orbit always has a natural manifold structure, modeled by the kernel of a suitable conditional expectation. When the spectrum of $a$…
In this paper we exploit the fact that a Cuntz C$^*$-algebra is a groupoid C$^*$-algebra to facilitate the study of non-self-adjoint subalgebras of $O_n$. The Cuntz groupoid is not principal and the spectral theorem for bimodules does not apply in full generality. We characterize the bimodules (over a natural masa) which are determined by their spectra in the Cuntz groupoid; these are exactly the ones which are invariant under the guage automorphisms and exactly the ones which are generated by the Cuntz partial isometries which they contain. We investigate analytic subalgebras of $O_n, n$ fini…
Let $P$ be a submonoid of a group $G$ and let $\mathcal{E}=(\mathcal{E}_p)_{p\in P}$ be a product system over $P$ with coefficient C*-algebra $A$. We show that the following C*-algebras are canonically isomorphic: the C*-envelope of the tensor algebra $\mathcal{T}_λ(\mathcal{E})^+$ of $\mathcal{E}$; the reduced cross sectional C*-algebra of the Fell bundle associated to the canonical coaction of $G$ on the covariance algebra $A\times_{\mathcal{E}}P$ of $\mathcal{E}$; and the C*-envelope of the cosystem obtained by restricting the canonical gauge coaction on $\mathcal{T}_λ(\mathcal{E})$ to the …
In this article we will study semigroupoids, and more specifically inverse semigroupoids. These are a common generalization to both inverse semigroups and groupoids, and provide a natural language on which several types of dynamical structures may be described. Moreover, this theory allows us to precisely compare and simultaneously generalize aspects of both the theories of inverse semigroups and groupoids. We begin by comparing and settling the differences between two notions of semigroupoids which appear in the literature (one by Tilson and another by Exel). We specialize this study to inver…
In this paper, we introduce C*-algebraic partial compact quantum groups, which are quantizations of topological groupoids with discrete object set and compact morphism spaces. These C*-algebraic partial compact quantum groups are generalisations of Hayashi's compact face algebras to the case where the object set can be infinite. They form the C*-algebraic counterpart of an algebraic theory of partial compact quantum groups developed in an earlier paper by the author and T. Timmermann, the correspondence between which will be dealt with in a separate paper. As an interesting example to illustra…
We consider three quantum algebras: the q-oscillator algebra, the Podles' sphere and the q-deformed enveloping algebra of $su(2).$ To each of these *-algebras we associate certain partial dynamical system and perform the "Mackey analysis" of *-representations developed in [SS]. As a result we get the description of "standard" irreducible *-representations. Further, for each of these examples we show the existence of a "$C^*$-envelope" which is canonically isomorphic to the covariance $C^*$-algebra of the partial dynamical system. Finally, for the q-oscillator algebra and the q-deformed $\cU(su…
We describe the $C^*$-algebra of an $E$-unitary or strongly 0-$E$-unitary inverse semigroup as the partial crossed product of a commutative $C^*$-algebra by the maximal group image of the inverse semigroup. We give a similar result for the $C^*$-algebra of the tight groupoid of an inverse semigroup. We also study conditions on a groupoid $C^*$-algebra to be Morita equivalent to a full crossed product of a commutative $C^*$-algebra with an inverse semigroup, generalizing results of Khoshkam and Skandalis for crossed products with groups.
