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Semigroup Forum, ISSN 0037-1912, 2019
Semigroup Forum, 2019 We extend Exel's ample tight groupoid construction to general locally compact \'etale groupoids in the Hausdorff case. Moreover, we show... 
Stone duality | Filters | Order structures | Patch topology | Inverse semigroups | Étale groupoids | Cosets | Local compactness | Domain theory
Journal Article
Advances in Mathematics, ISSN 0001-8708, 09/2013, Volume 244, pp. 117 - 170
A pseudogroup is a complete infinitely distributive inverse monoid. Such inverse monoids bear the same relationship to classical pseudogroups of... 
Inverse semigroups | Locales | Topos theory | Étale topological groupoids | MATHEMATICS | INVERSE-SEMIGROUPS | ALGEBRAS | EXTENSIONS | COMPLETIONS | TILINGS | POLYCYCLIC MONOIDS | STONE DUALITY | Etale topological groupoids | Algebra
Journal Article
SEMIGROUP FORUM, ISSN 0037-1912, 04/2014, Volume 88, Issue 2, pp. 433 - 452
We prove that the full C (au)-algebra of a second-countable, Hausdorff, ,tale, amenable groupoid is simple if and only if the groupoid is both topologically... 
MATHEMATICS | Groupoid | INVERSE SEMIGROUP | CROSSED PRODUCT | DYNAMICAL-SYSTEMS | Simple C-algebra | Leavitt path algebra | C-ASTERISK-ALGEBRAS | Groupoid algebra | ENDOMORPHISMS | HIGHER-RANK GRAPHS | ETALE GROUPOIDS
Journal Article
Journal of Functional Analysis, ISSN 0022-1236, 10/2018, Volume 275, Issue 7, pp. 1756 - 1807
We develop a generalization of quantitative -theory, which we call controlled -theory. It is powerful enough to study the -theory of crossed product of... 
Etale groupoids | Coarse Baum–Connes conjecture | K-theory and K-homology | Baum–Connes conjecture | MATHEMATICS | NOVIKOV-CONJECTURE | Coarse Baum-Connes conjecture | Baum-Connes conjecture | BANACH-ALGEBRAS | Algebra
Journal Article
Advances in Mathematics, ISSN 0001-8708, 2007, Volume 208, Issue 1, pp. 147 - 209
We establish close and previously unknown relations between quantales and groupoids. In particular, to each étale groupoid, either localic or topological,... 
Inverse semigroup | Pseudogroup | Topological groupoid | Localic groupoid | Quantale | Étale groupoid | MATHEMATICS | ALGEBRAS | inverse semigroup | topological groupoid | Etale groupoid | QUANTIZATION | pseudogroup | quantale | localic groupoid
Journal Article
Advances in Mathematics, ISSN 0001-8708, 11/2016, Volume 303, pp. 502 - 548
Two conjectures about homology groups, -groups and topological full groups of minimal étale groupoids on Cantor sets are formulated. We verify these... 
Shifts of finite type | Higman–Thompson groups | Etale groupoids | Topological full groups | CANTOR MINIMAL SYSTEMS | DIMENSIONAL THOMPSON GROUPS | SPACES | PRESENTATIONS | MATHEMATICS | ALGEBRAS | Higman-Thompson groups | GROUPS NV | HOMOLOGY | ORBIT EQUIVALENCE
Journal Article
Journal of Algebra, ISSN 0021-8693, 01/2019, Volume 518, pp. 412 - 439
Work of Jean Renault shows that, for topologically principal étale groupoids, a diagonal-preserving isomorphism of reduced -algebras yields an isomorphism of... 
Groupoid algebra | Diagonal-preserving isomorphism | Étale groupoid | MATHEMATICS | STEINBERG ALGEBRAS | SIMPLICITY | Etale groupoid | LEAVITT PATH ALGEBRAS | Algebra
Journal Article
Journal of Pure and Applied Algebra, ISSN 0022-4049, 08/2015, Volume 219, Issue 8, pp. 3089 - 3109
We provide three functorial extensions of the equivalence between localic étale groupoids and their quantales. The main result is a biequivalence between the... 
MATHEMATICS | MATHEMATICS, APPLIED | CLASSICAL PSEUDOGROUPS | SPACES | QUANTUM | C-STAR-ALGEBRAS | QUANTIZATION | ETALE GROUPOIDS
Journal Article
International Journal of Algebra and Computation, ISSN 0218-1967, 03/2016, Volume 26, Issue 2, pp. 375 - 397
We study growth and complexity of étale groupoids in relation to growth of their convolution algebras. As an application, we construct simple finitely... 
étale groupoids | Gelfand-Kirillov dimension | Growth of algebras | MATHEMATICS | etale groupoids | SEQUENCES | COMPLEXITY | SELF-SIMILAR GROUPS | SYSTEMS | OPERATOR-ALGEBRAS
Journal Article
Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 01/2019, Volume 469, Issue 2, pp. 1088 - 1110
We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using... 
Continuous orbit equivalence | Flow equivalence | Leavitt path algebras | algebras | Diagonal-preserving isomorphisms | Étale groupoids | Graph C | MATHEMATICS, APPLIED | SPACES | MATHEMATICS | CUNTZ-KRIEGER ALGEBRAS | Etale groupoids | TOPOLOGICAL MARKOV SHIFTS | C-STAR-ALGEBRAS | CHAINS | Graph C-algebras | Markov processes | Algebra
Journal Article
Journal of Algebra, ISSN 0021-8693, 03/2018, Volume 498, pp. 197 - 210
We study the projections of an arbitrary stably Gelfand quantale and show that each projection determines a pseudogroup (and a corresponding localic étale... 
Étale groupoids | Stably Gelfand quantales | Pseudogroups | Inverse quantal frames | MATHEMATICS | C-ASTERISK-ALGEBRAS | Etale groupoids | QUANTIZATION
Journal Article
Topology and its Applications, ISSN 0166-8641, 2012, Volume 159, Issue 3, pp. 756 - 786
We define a 2-category structure ( ) on the category of reduced complex orbifold atlases. We construct a 2-functor from ( ) to the 2-category ( ) of proper... 
2-Categories and 2-functors | Morita equivalences | Reduced complex orbifolds | Proper étale effective groupoid objects over complex manifolds | Secondary | Primary | MATHEMATICS | MATHEMATICS, APPLIED | COHOMOLOGY | Proper etale effective groupoid objects over complex manifolds | STACKS
Journal Article
Proceedings of the Edinburgh Mathematical Society, ISSN 0013-0915, 10/2010, Volume 53, Issue 3, pp. 765 - 785
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... 
translation groupoid | coarse geometry | complete inverse semigroup | topological etale groupoid | universal groupoid | localic groupoid | MATHEMATICS | Topology | Inverse | Representations | Mathematical analysis | Translations | Monoids
Journal Article
Journal of the Australian Mathematical Society, ISSN 1446-7887, 04/2013, Volume 94, Issue 2, pp. 234 - 256
Paterson showed how to construct an etale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by... 
inverse semigroups | Étale groupoids | filters | MATHEMATICS | Etale groupoids | TILINGS
Journal Article
Journal of Noncommutative Geometry, ISSN 1661-6952, 2012, Volume 6, Issue 2, pp. 199 - 247
It is well known that etale groupoids are closely related to inverse semigroups. In particular, it has recently been shown that there is a non-functorial... 
Topological groupoid | Lie groupoid | Quantale | Inverse semigroup | Pseudogroup | Localic groupoid | Open groupoid | MATHEMATICS | MATHEMATICS, APPLIED | inverse semigroup | topological groupoid | open groupoid | PHYSICS, MATHEMATICAL | pseudogroup | ETALE GROUPOIDS | localic groupoid
Journal Article
Semigroup Forum, ISSN 0037-1912, 10/2017, Volume 95, Issue 2, pp. 321 - 344
We investigate recent uniqueness theorems for reduced $$C^*$$ C ∗ -algebras of Hausdorff étale groupoids in the context of inverse semigroups. In many cases... 
Inverse semigroups | Mathematics | C^$$ C ∗ -algebras | Algebra | étale groupoids | algebras | MATHEMATICS | etale groupoids | SEMILATTICES | SIMPLICITY | C-algebras | C-ASTERISK-ALGEBRAS | TALE GROUPOIDS
Journal Article
Journal of Pure and Applied Algebra, ISSN 0022-4049, 06/2019, Volume 223, Issue 6, pp. 2474 - 2488
In this paper we give some sufficient and some necessary conditions for an étale groupoid algebra to be a prime ring. As an application we recover the known... 
Inverse semigroups | Étale groupoids | Groupoid algebras | Prime rings | Leavitt path algebras | MATHEMATICS | SEMIPRIMITIVITY | MATHEMATICS, APPLIED | STEINBERG ALGEBRAS | SIMPLICITY | Etale groupoids
Journal Article
Semigroup Forum, ISSN 0037-1912, 2/2016, Volume 92, Issue 1, pp. 274 - 303
In this work we present algebraic conditions on an inverse semigroup $$\mathcal {S}$$ S (with zero) which imply that its associated tight groupoid $$\mathcal... 
Ultra-filter | Groupoid | Algebra | Tight character | Tight filter | Groupoid C-algebra | Inverse semigroup | Semi-lattice | Mathematics | MATHEMATICS | C-ASTERISK-ALGEBRAS | CONSTRUCTION | ETALE GROUPOIDS
Journal Article
Journal of the Australian Mathematical Society, ISSN 1446-7887, 06/2018, Volume 104, Issue 3, pp. 403 - 411
The author has previously associated to each commutative ring with unit R and etale groupoid G with locally compact, Hausdorff and totally disconnected unit... 
inverse semigroup algebras | chain conditions | groupoid algebras | Leavitt path algebras | étale groupoids | GRAPH | MATHEMATICS | REPRESENTATION-THEORY | etale groupoids | STEINBERG ALGEBRAS | SIMPLICITY | RINGS
Journal Article
Journal of the Australian Mathematical Society, ISSN 1446-7887, 06/2014, Volume 97, Issue 3, pp. 418 - 429
The author has previously associated to each commutative ring with unit k and etale groupoid G with locally compact, Hausdorff, totally disconnected unit space... 
groupoid algebras | étale groupoids | sheaves | MATHEMATICS | MORITA EQUIVALENCE | etale groupoids | RINGS
Journal Article
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