Inverse semigroups and combinatorial C*-algebras

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 semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case. *Partially supported by CNPq.     

Groupoid cohomology and extensions

We show that Haefliger's cohomology for étale 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.

     

Topologies on measured groupoids

If an analytic Borel group G has a quasiinvariant measure, it is known that G is actually a locally compact group with the original Borel structure being generated by the topology and the original measure being equivalent to Haar measure. In this paper a variation is given on the known proof which then extends to show that an analytic measured groupoid has a σ-compact, and also a locally compact, inessential reduction which is a topological groupoid. In the σ-compact case, it is proved that every “almost” homomorphism agrees a.e. with a (strict) homomorphism. Also, the topology is used to show that every measured groupoid has a complete countable section ¦7¦ and that every locally compact equivalence relation has a complete transversal ¦3¦. These are further used to show that some results of Feldman et al. ¦7¦ apply in general and that a locally compact groupoid with (continuous) Haar system has sufficiently many non-singular Borel G-sets provided that the orbit measures are atom-free ¦23¦.     

I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type

We introduce a notion of I-factorial quantum torsor, which consists of an integrable ergodic action of a locally compact quantum group on a type I-factor such that also the crossed product is a type I-factor. We show that any such I-factorial quantum torsor is at the same time a I-factorial quantum torsor for the dual locally compact quantum group, in such a way that the construction is involutive. As a motivating example, we show that quantized compact semisimple Lie groups, when amplified via a crossed product construction with the function algebra on the associated weight lattice, admit I-factorial quantum torsors, and give an explicit realization of the dual quantum torsor in terms of a deformed Heisenberg algebra for the Borel part of a quantized universal enveloping algebra.     

An asymmetric Ellis theorem

In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual, Tk, to obtain the following asymmetric Ellis theorem which applies to the example above:Whenever (X,⋅,T) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both (X,⋅,T) and (X,⋅,Tk), and inversion is a homeomorphism between (X,T) and (X,Tk).This generalizes the classical Ellis theorem, because T=Tk when (X,T) is locally compact Hausdorff.     

Square Integrable Representation of Groupoids

A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur＇s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle H = （Hu）u∈G^0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle （G^0, （Hu）,μ）, then dimHu = 1 （u ∈ G^0）. Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.     

The Primitive Ideals of some Étale Groupoid <Emphasis Type="Italic">C</Emphasis><Superscript>∗</Superscript>-Algebras

We consider the Deaconu–Renault groupoid of an action of a finitely generated free abelian monoid by local homeomorphisms of a locally compact Hausdorff space. We catalogue the primitive ideals of the associated groupoid C ^?-algebra. For a special class of actions we describe the Jacobson topology.     

The Furstenberg Boundary of Groupoids

Siberian Mathematical Journal - Let $ {\mathcal{G}} $ be a locally compact groupoid. We show that there is a&nbsp;one-to-one correspondence between $ {\mathcal{G}} $ -spaces and the groupoid...     

Orbit structure and countable sections for actions of continuous groups

It is shown that if a second countable locally compact group G acts nonsingularly on an analytic measure space (S, μ), then there is a Borel subset E ? S such that EG is conull in S and each sG ∩ E is countable. It follows that the measure groupoid constructed from the equivalence relation s ～ sg on E may be simply described in terms of the measure groupoid made from the action of some countable group. Some simplifications are made in Mackey's theory of measure groupoids. A natural notion of “approximate finiteness” (AF) is introduced for nonsingular actions of G, and results are developed parallel to those for countable groups; several classes of examples arising naturally are shown to be AF. Results on “skew product” group actions are obtained, generalizing the countable case, and partially answering a question of Mackey. We also show that a group-measure space factor obtained from a continuous group action is isomorphic (as a von Neumann algebra) to one obtained from a discrete group action.     

Locally Hausdorff tight groupoids generalised

We extend Exel’s ample tight groupoid construction to non-ample groupoids, even in the general locally Hausdorff case.

     

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space which is totally disconnected, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{?}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.     

The primitive dual space of [FC]− groups

We show that if G is a σ compact locally compact group with relatively compact conjugacy classes, then the enveloping C¹-algebra $\text{C}{}^1\text{(G)}$ has a Hausdorff primitive ideal space. We also discuss some open problems and a partial converse result.     

Symmetric nonself-adjoint operators in an enveloping algebra

Nelson and Stinespring proved that in any unitary representation of a Lie group with compact Lie algebra the representation of Hermitian elements in the enveloping algebra are essentially self-adjoint. If the Lie algebra is noncompact, we construct in its enveloping algebra a Hermitian element u such that in any locally faithful unitary representation the representative of u has no self-adjoint extension.     

Ordinary semicascades and their ergodic properties

A relationship is considered between ergodic properties of a discrete dynamical system on a compact metric space Ω and characteristics of companion algebro-topological objects, namely, the Ellis enveloping semigroup E, the Köhler enveloping operator semigroup Γ, and the semigroup G being the closure of the convex hull of Γ in the weak-star topology on the operator space EndC*(Ω). The main results are formulated for ordinary (having metrizable semigroup E) semicascades and for tame dynamical systems determined by the condition cardE ? c. A classification of compact semicascades in terms of topological properties of the semigroups specified above is given.     

Zappa–Szép product groupoids and $$C^*$$-blends

We study the external and internal Zappa–Szép product of topological groupoids. We show that under natural continuity assumptions the Zappa–Szép product groupoid is étale if and only if the individual groupoids are étale. In our main result we show that the \(C^*\)-algebra of a locally compact Hausdorff étale Zappa–Szép product groupoid is a \(C^*\)-blend, in the sense of Exel, of the individual groupoid \(C^*\)-algebras. We finish with some examples, including groupoids built from \(*\)-commuting endomorphisms, and skew product groupoids.     

Proper groupoids and momentum maps: Linearization, affinity, and convexity

We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie group action). In the case of proper (quasi-)symplectic groupoids, the orbit space admits a natural integral affine structure, which makes it into an affine orbifold with locally convex polyhedral boundary, and the local structure near each boundary point is isomorphic to that of a Weyl chamber of a compact Lie group. We then apply these results to the study of momentum maps of Hamiltonian actions of proper (quasi-)symplectic groupoids, and show that these momentum maps preserve natural transverse affine structures with local convexity properties. Many convexity theorems in the literature can be recovered from this last statement and some elementary results about affine maps.     

Weakly Linearly Compact Topological Abelian Groups

In this paper, we study linearly topological groups. We introduce the notion of a weakly linearly compact group, which generalizes the notion of a weakly separable group, and examine the main properties of such groups. For weakly linearly compact groups, we construct the character theory and present an algebraic characterization of some classes of such groups. Some well-known theorems for periodic Abelian groups are generalized for the case of linearly discrete, topological Abelian groups; for linearly compact and linearly discrete topological Abelian groups, we also construct the character theory and study some important properties of linearly discrete groups. For linearly discrete, topological Abelian groups, we analyze the splittability condition (Theorem 3.12) and present the characteristic condition of decomposability of a discrete group G into the direct sum of rank-1 groups. We also present an algebraic characterization of linearly compact groups. We introduce the notion of a weakly linearly compact, topological Abelian group, which generalizes the notion of a weakly separable Abelian group, and examine some properties of such groups. These groups are a generalization of fibrous Abelian groups introduced by Vilenkin. We give an algebraic characterization of divisible, weakly locally compact Abelian groups that do not contain nonzero elements of finite order (Proposition 7.9). For weakly locally compact Abelian groups, we construct universal groups.     

Projektive Inzidenzgruppoide und Fastalgebren

In this paper we extend the notion of an incidence group to that of a groupoid. The concept of near-algebras, introduced below, enables us to describe Desarguesian incidence groupoids algebraically and to generalize most of the essential results about projective incidence groups with regard to groupoids.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag     

La conjecture de Baum–Connes pour les feuilletages moyennables

We show, using the construction of Higson and Kasparov, that the Baum–Connes Conjecture holds for foliations whose holonomy groupoid is Hausdorff and amenable. More generally, for every locally compact, -compact and Hausdorff groupoid G acting continuously and isometrically on a continuous field of affine Euclidean spaces, the Baum–Connes conjecture with coefficients is an isomorphism, and G amenable in K-theory. In addition, we show that C^*(G) satisfies the Universal Coefficient Theorem.     

Towards a 2-dimensional notion of holonomy

Previous work (Pradines, C. R. Acad. Sci. Paris 263 (1966) 907; Aof and Brown, Topology Appl. 47 (1992) 97) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involves the 2-dimensional information.