For $p\in (1,\infty)$, we study representations of étale groupoids on $L^{p}$-spaces. Our main result is a generalization of Renault's disintegration theorem for representations of étale groupoids on Hilbert spaces. We establish a correspondence between $L^{p}$-representations of an étale groupoid $G$, contractive $L^{p}$-representations of $C_{c}(G)$, and tight regular $L^{p}$-representations of any countable inverse semigroup of open slices of $G$ that is a basis for the topology of $G$. We define analogs $F^{p}(G)$ and $F_{\mathrm{red}}^{p}(G)$ of the full and reduced groupoid C*-algebras using representations on $L^{p}$-spaces. As a consequence of our main result, we deduce that every contractive representation of $F^{p}(G)$ or $F_{\mathrm{red}}^{p}(G)$ is automatically completely contractive. Examples of our construction include the following natural families of Banach algebras: discrete group $L^{p}$-operator algebras, the analogs of Cuntz algebras on $L^{p}$-spaces, and the analogs of AF-algebras on $L^{p}$-spaces. Our results yield new information about these objects: their matricially normed structure is uniquely determined. More generally, groupoid $L^{p}$-operator algebras provide analogs of several families of classical C*-algebras, such as Cuntz-Krieger C*-algebras, tiling C*-algebras, and graph C*-algebras.
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.
Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of T such that G:=E/T is a principal groupoid with Haar system λ. The twisted groupoid C*-algebra C*(E;G,λ) is a quotient of the C*-algebra of E obtained by completing the space of T-equivariant functions on E. We show that C*(E;G,λ) is postliminal if and only if the orbit space of G is T_0 and that C*(E;G, λ) is liminal if and only if the orbit space is T_1. We also show that C*(E;G, λ) has bounded trace if and only if G is integrable and that C*(E;G, λ) is a Fell algebra if and only if G is Cartan. Let \…
We first describe a Rieffel induction system for groupoid crossed products. We then use this induction system to show that, given a regular groupoid $G$ and a dynamical system $(A,G,α)$, every irreducible representation of $A\rtimes G$ is induced from a representation of the group crossed product $A(u)\rtimes S_u$ where $u\in G\unit$, $A(u)$ is a fibre of $A$, and $S_u$ is a stabilizer subgroup of $G$.
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…
If $G$ is a locally compact groupoid with a Haar system $λ$, then a positive definite function $p$ on $G$ has a form $p(x)=< L(x)ξ(d(x)),ξ(r(x))>$, where $L$ is a representation of $G$ on a Hilbert bundle ${\h}=(G^0,\{H_u\},μ)$, $μ$ is a quasi invariant measure on $G^0$ and $ξ\in L^{\infty}(G^0,{\h})$. [10]. In this paper firt we prove that if $μ$ is a quasi invariant ergodic measure on $G^0$, then two corresponding representations of $G$ and $C_c(G)$ are irreducible in the same time. Then by using the theory of positive linear functionals on $C^*(G)$ we show that when $μ$ is an ergodic quasi …
In a series of papers, we have shown that from the representatio theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study continuous representations of compact groupoids. We show that irreducible representations have finite dimensional fibres. We prove the Schur's lemma, Gelfand-Raikov theorem and Peter-Weyl theorem for compact groupoids.
If $G$ is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible.
Let G be a second-countable locally-compact Hausdorff groupoid with a Haar system, and let {x_n} be a sequence in the unit space of G. We show that the notions of strength of convergence of {x_n} in the orbit space and measure-theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representation of the groupoid C*-algebra.
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.
This dissertation concerns the classification of groupoid and higher-rank graph C*-algebras and has two main components. Firstly, for a groupoid it is shown that the notions of strength of convergence in the orbit space and measure-theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representations of the groupoid C*-algebra. Examples of directed graphs are given, showing how to determine the multiplicity numbers associated to various sequences of induced representations of the directed graph C*-algebras. The second …
We extend the symbol calculus and study the limit operator theory for $σ$-compact, étale and amenable groupoids, in the Hilbert space case. This approach not only unifies various existing results which include the cases of exact groups and discrete metric spaces with Property A, but also establish new limit operator theories for group/groupoid actions and uniform Roe algebras of groupoids. In the process, we extend a monumental result by Exel, Nistor and Prudhon, showing that the invertibility of an element in the groupoid $C^*$-algebra of a $σ$-compact amenable groupoid with a Haar system is …
Norm upper-semicontinuity of functions supported on open abelian isotropy in étale groupoids (a corrigendum to "Reconstruction of groupoids and C*-rigidity of dynamical systems," Adv. Math 390 (2021), 107923)
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 second-countable, Hausdorff, étale, amenable groupoid G with compact unit space, we show that an element a in C*(G) is invertible if and only if λ_x(a) is invertible for every x in the unit space of G, where λ_x refers to the "regular representation" of C*(G) on l_2(G_x). We also prove that, for every a in C*(G), there exists some x in G^{(0)} such that ||a|| = ||λ_x(a)||.
This is Part II in our multi-part series of papers developing the theory of a subclass of locally compact quantum groupoids ("quantum groupoids of separable type"), based on the purely algebraic notion of weak multiplier Hopf algebras. The definition was given in Part I. The existence of a certain canonical idempotent element $E$ plays a central role. In this Part II, we develop the main theory, discussing the structure of our quantum groupoids. We will construct from the defining axioms the right/left regular representations and the antipode map.
Given an ample groupoid $G$ with compact unit space, we study the canonical representation of the topological full group $[[G]]$ in the full groupoid $C^*$-algebra $C^*(G)$. In particular, we show that the image of this representation generates $C^*(G)$ if and only if $C^*(G)$ admits no tracial state. The techniques that we use include the notion of groups covering groupoids. As an application, we provide sufficient conditions for $C^*$-simplicity of certain topological full groups, including those associated with topologically free and minimal actions of non-amenable and countable groups on t…
The paper contains a description of the KMS weights for the one-parameter action on the reduced C*-algebra of a second countable locally compact Hausdorff etale groupoid, arising from a continuous real valued homomorphism satisfying two conditions. The results are subsequently applied to identify the KMS weights for the gauge action on a simple graph algebra. The von Neumann algebra generated by the GNS-representation of an extremal KMS weight is a factor, and tools are developed to determine its type. The paper concludes with three examples to illustrate the results.
We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the C*-algebra associated to non-Hausdorff étale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that C*-algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coef…
The study of different types of ideals in non self-adjoint operator algebras has been a topic of recent research. This study focuses on principal ideals in subalgebras of groupoid C*-algebras. An ideal is said to be principal if it is generated by a single element of the algebra. We look at subalgebras of r-discrete principal groupoid C*-algebras and prove that these algebras are principal ideal algebras. Regular canonical subalgebras of almost finite C*-algebras have digraph algebras as their building blocks. The spectrum of almost finite C*-algebras has the structure of an r-discrete princip…
We introduce the $C^*$-algebra $C^*(κ)$ generated by the Koopman representation $κ$ of an étale groupoid $G$ acting on a measure space $(X,μ)$. We prove that for a level transitive self-similar action $(G,E)$ with $E$ finite and $|uE^1|$ constant, there is an invariant measure $ν$ on $X=E^\infty$ and that $C^*(κ)$ is residually finite-dimensional with a normalized self-similar trace. We also discus $p$-fold similarities of Hilbert spaces in connection to representations of the graph algebra $C^*(E)$ and self-similar representations of $G$ in connection to the Cuntz-Pimsner algebra $C^*(G,E)$.
Categories of paths are a generalization of several kinds of oriented discrete data that have been used to construct $C^*$-algebras. The techniques introduced to study these constructions apply almost verbatim to the more general situation of left cancellative small categories. We develop this theory and derive the structure of the $C^*$-algebras in the most general situation. We analyze the regular representation, and the Wiener-Hopf algebra in the case of a subcategory of a groupoid.
We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author's Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal)…
We introduce the notion of wide representation of an inverse semigroup and prove that with a suitably defined topology there is a space of germs of such a representation which has the structure of an etale groupoid. This gives an elegant description of Paterson's universal groupoid and of the translation groupoid of Skandalis, Tu, and Yu. In addition we characterize the inverse semigroups that arise from groupoids, leading to a precise bijection between the class of etale groupoids and the class of complete and infinitely distributive inverse monoids equipped with suitable representations, and…
We initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of Braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-a…
We characterize the groupoids for which an operator is Fredholm if, and only if, its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called {\em Fredholm}. Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that a…
We fix a path model for the space of filters of the inverse semigroup $\mathcal{S}_Λ$ associated to a left cancellative small category $Λ$. Then, we compute its tight groupoid, thus giving a representation of its $C^*$-algebra as a (full) groupoid algebra. Using it, we characterize when these algebras are simple. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses.
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.
Normal elements (or multipliers) of the C* algebra of a certain class of locally compact groupoids admit a natural faithful representation as normal operators on the $L^2$-space of a dense orbit of the groupoid. We prove norm estimates on the product between elements of the functional calculus of these operators and multiplication operators, subject to suitable restrictions expressed in terms of the orbit structure of the groupoid. As a consequence, one gets uniform estimates on the local behavior of the evolution group of the operators.
We study bounded operators defined in terms of the regular representations of the $C^*$-algebra of an amenable, Hausdorff, second countable locally compact groupoid endowed with a continuous $2$-cocycle. We concentrate on spectral quantities associated to natural quotients of this twisted algebra, such as the essential spectrum, the essential numerical range, and Fredholm properties. We obtain decompositions for the regular representations associated to units of the groupoid belonging to a free locally closed orbit, in terms of spectral quantities attached to points (or orbits) in the boundary…
We treat spectral problems by twisted groupoid methods. To Hausdorff locally compact groupoids endowed with a continuous $2$-cocycle one associates the reduced twisted groupoid $C^*$-algebra. Elements (or multipliers) of this algebra admit natural Hilbert space representations. We show the relevance of the orbit closure structure of the unit space of the groupoid in dealing with spectra, norms, numerical ranges and $ε$-pseudospectra of the resulting operators. As an example, we treat a class of pseudo-differential operators introduced recently, associated to group actions. We also prove a Deco…
We extend the definitions and main properties of graded extensions to the category of locally compact groupoids endowed with involutions. We introduce Real Čech cohomology, which is an equivariant-like cohomology theory suitable for the context of groupoids with involutions. The Picard group of such a groupoid is discussed and is given a cohomological picture. Eventually, we generalize Crainic's result, about the differential cohomology of a proper Lie groupoid with coefficients in a given representation, to the topological case.
Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal étale groupoid associated to $S$ by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal $C^*$-algebra. It provides a convenient topological framework for understanding the structure of $KS$, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality. Using …
Let G be a locally compact, Hausdorff groupoid in which s is a local homeomorphism and the unit space is totally disconnected. Assume there is a continuous cocycle c from G into a discrete group $Γ$. We show that the collection A(G) of locally-constant, compactly supported functions on G is a dense *-subalgebra of C_c(G) and that it is universal for algebraic representations of the collection of compact open bisections of G. We also show that if G is the groupoid associated to a row-finite graph or k-graph with no sources, then A(G) is isomorphic to the associated Leavitt path algebra or Kumji…
In this paper we establish a duality between etale Lie groupoids and a class of non-necessarily commutative algebras with a Hopf algebroid structure. For any etale Lie groupoid G over a manifold M, the groupoid algebra C_c(G) of smooth functions with compact support on G has a natural coalgebra structure over C_c(M) which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over C_c(M) we construct the associated spectral etale Lie groupoid G_sp(A) over M such that G_sp(C_c(G)) is naturally isomorphic to G. Both these constructions are functorial, and C_c is fully faithful left…
Persson's formula expresses the infimum of the essential spectrum of a suitable self-adjoint Schrödinger operator in $R^n$ in terms of the lower spectral points of a family of restrictions of the operator to complements of relatively compact subsets. It has been extended to other situations. We present an approach based on $C^*$-algebras associated to étale groupoids. In such a setting there are intrinsic versions, referring to self-adjoint elements of the groupoid $C^*$-algebras. When representations in Hilbert spaces are considered, the results not always involve only the essential spectrum.…
We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term "tight". These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the "tight spectrum", which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigr…
We use a groupoid model for the spin algebra to introduce boundary conditions on quantum spin systems via a Poisson point process representation. We can describe KMS states of quantum systems by means of a set of equations resembling the standard DLR equations of classical statistical mechanics. We introduce a notion of quantum specification which recovers the classical DLR measures in the particular case of classical interactions. Our results are in the same direction as those obtained recently by Cha, Naaijkens, and Nachtergaele, differently somehow from the predicted by Fannes and Werner.
We consider the algebra $\dotΣ(\mathcal L)$ generated by the inner-limit derivations over the ${\rm GICAR}$ algebra of a fermion gas populating an aperiodic Delone set $\mathcal L$. Under standard physical assumptions such as finite interaction range, Galilean invariance and continuity with respect to the aperiodic lattice, we demonstrate that the image of $\dot Σ(\mathcal L)$ through the Fock representation can be completed to a groupoid-solvable pro-$C^\ast$-algebra. Our result is the first step towards unlocking the $K$-theoretic tools available for separable $C^\ast$-algebra for applicatio…
