Partial actions of groups and actions of inverse semigroups

Exel, Ruy

doi:10.1090/s0002-9939-98-04575-4

Cited by 185 publications

(170 citation statements)

References 11 publications

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“…The proofs of Theorem 5.4 and Corollary 5.7 still work for non-saturated Fell bundles (even over inverse semigroups). Alternatively, we may replace our nonsaturated Fell bundle over G by a saturated Fell bundle over an inverse semigroup associated to G, just as for partial actions (see [11]). This does not change the section C * -algebra, and afterwards Theorem 5.4 applies literally.…”

Section: Construction Of the Fell Bundlementioning

confidence: 99%

Inverse semigroup actions on groupoids

Buss¹,

Meyer²

2017

Rocky Mountain J. Math.

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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 \'etale 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 \'etale 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 does not generalise to non-Hausdorff groupoids.Comment: 59 pages. Proofreading with minor changes. Version accepted for publication at Rocky Mountain Journa

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Section: Construction Of the Fell Bundlementioning

confidence: 99%

Inverse semigroup actions on groupoids

Buss¹,

Meyer²

2017

Rocky Mountain J. Math.

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“…As it is was mentioned in many occasions, the formal concept of a partial action (in the sense we are using it) appeared in the theory C * -algebras (see [146,149,151,232]), permitting one to endow relevant classes of C * -algebras with a general structure of a partial crossed product [147,148,150,160,264] (see also [155]), and promptly stimulating further use and discussions in the area [1,3,4,150,152,157,158,264,[271][272][273]. Subsequent C * -algebraic and topological developments on partial actions were made in [5,6,11,60,93,112,141,159,163,222,233].…”

Section: Mathematics Subject Classificationmentioning

confidence: 99%

“…The first answer was given by Exel [151,Proposition 4.1] relating it to the important concept of a partial representation: a map α : G → I(X ) gives a partial action if and only if for all g, h ∈ G we have…”

Section: Mathematics Subject Classificationmentioning

confidence: 99%

“…The first algebraic results on the subject appeared in [89,91,117,120,151,204,205,275,276], including the first algebraic application for tiling semigroups in [204] (with further use in [283]), for E-unitary inverse semigroups in [205], to inverse semigroups and F-inverse monoids in [275] and to inverse monoids of Möbius type in [91]. Independently from Exel's definition of a partial action, Coulbois [110] used a more restrictive notion, called a pre-action, to deal with the Ribes-Zalesski property (R Z n ) of groups from model theoretic point of view (see also [111]).…”

Section: Mathematics Subject Classificationmentioning

confidence: 99%

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Recent developments around partial actions

Dokuchaev

2018

São Paulo J. Math. Sci.

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We give an overview of publications on partial actions and related concepts, paying main attention to some recent developments on diverse aspects of the theory, such as partial actions of semigroups, of Hopf algebras and groupoids, the globalization problem for partial actions, Morita theory of partial actions, twisted partial actions, partial projective representations and the Schur multiplier, cohomology theories related to partial actions, Galois theoretic results, ring theoretic properties and ideals of partial crossed products. Among the applications we consider in more detail the case of the Carlsen-Matsumoto C *-algebra related to an arbitrary subshift, but also mention many others. The total number of publications directly related to partial actions and partial representations is more than 130, so that it is impossible even to describe briefly the content of all of them within the constraints of the present survey. Thus, the majority of them are only cited with respects to specific topics, trying to give an idea about the involved matter. In order to complete the picture, we refer the reader to a recent book by Ruy Exel, to our previous surveys, as well as to those by other authors.

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“…Note that forms a subsemigroup of the inverse semigroup of partially defined bijective maps X (see, e.g., [ 36 ]). More explicitly, the composition of any two partial translations , denoted by , is defined to be the partial translation satisfying and for any .…”

Section: Amenable Metric Spacesmentioning

confidence: 99%

Amenability of coarse spaces and $$\mathbb {K}$$ K -algebras

Ara

Lledó

et al. 2017

Bull. Math. Sci.

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In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite Communicated by Efim Zelmanov. extended metric spaces and for general algebras over fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra.

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Partial actions of groups and actions of inverse semigroups

Cited by 185 publications

References 11 publications

Inverse semigroup actions on groupoids

Inverse semigroup actions on groupoids

Recent developments around partial actions

Amenability of coarse spaces and $$\mathbb {K}$$ K -algebras

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