Quotients of Étale Groupoids and the Abelianizations of Groupoid C*-Algebras

Komura, Fuyuta

doi:10.1017/s1446788720000154

Cited by 3 publications

(3 citation statements)

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“…is an étale groupoid with respect to the quotient topology. Moreover, the quotient map is a local homeomorphism (see [1,Section 3.1] for these facts).…”

Section: éTale Groupoidsmentioning

confidence: 99%

“…For an étale groupoid G, the author of [1] constructed the étale abelian group bundle G ab . We briefly recall the definition of G ab .…”

Section: Proposition 21 Let G and H Be éTale Groupoids A Groupoid Hom...mentioning

confidence: 99%

“…See [4] for the definition of the universal groupoid C*-algebras. THEOREM 2.3 [1,Theorem 4.12]. Let G be an étale groupoid such that G (0) is a locally compact Hausdorff space with respect to the relative topology of G. Then C * (G ab ) is the abelianization of C * (G) as a C*-algebra.…”

Section: Proposition 21 Let G and H Be éTale Groupoids A Groupoid Hom...mentioning

confidence: 99%

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Invariant Sets and Normal Subgroupoids of Universal Étale Groupoids Induced by Congruences of Inverse Semigroups

Komura

2021

J. Aust. Math. Soc.

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For a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept for considering quotients of inverse semigroups. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quotient of the original universal groupoid. Finally we compute invariant sets and normal subgroupoids induced by special congruences including abelianization.

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“…is an étale groupoid with respect to the quotient topology. Moreover, the quotient map is a local homeomorphism (see [1,Section 3.1] for these facts).…”

Section: éTale Groupoidsmentioning

confidence: 99%

“…For an étale groupoid G, the author of [1] constructed the étale abelian group bundle G ab . We briefly recall the definition of G ab .…”

Section: Proposition 21 Let G and H Be éTale Groupoids A Groupoid Hom...mentioning

confidence: 99%

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Invariant Sets and Normal Subgroupoids of Universal Étale Groupoids Induced by Congruences of Inverse Semigroups

Komura

2021

J. Aust. Math. Soc.

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Residually linear abstract groupoids

Draoui,

Choulli,

Mouanis

2024

Известия Российской академии наук. Серия математическая

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We introduce the notion of residually linear groupoids. We characterize this class in analogy with the group-theoretic setting. Various properties are proved and a relationship with residual finiteness is investigated. From a categorical point of view, our approach extends some well-known results in the theory of discrete groups, due mainly to Mal'cev and Menal. Finally, as an application, we show that the character groupoid of the Hopf algebroid of representative functions of a transitive groupoid is always residually linear. Bibliography: 24 titles.

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