On zigzag maps and the path category of an inverse semigroup

Donsig, Allan P.; Gensler, Jennifer; King, Hannah; Milan, David; Wdowinski, Ronen

doi:10.1007/s00233-019-10031-2

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…Moreover, since E * is singly aligned, then so is E * ⋊ ϕ G by [2,Proposition 3.12(ii)]. Thus, by [5,Theorem 3.2],…”

Section: Zappa-szép Products Of Lcsc Categoriesmentioning

confidence: 92%

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The tight groupoid of the inverse semigroups of left cancellative small categories

Ortega¹,

Pardo²

2020

Trans. Amer. Math. Soc.

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We fix a path model for the space of filters of the inverse semigroup 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. 2010 Mathematics Subject Classification. Primary: 46L05; Secondary: 46L80, 46L55, 20L05. . α ∨ β := {the minimal extensions of α and β} .Notice that if α ∨ β = ∅ then α ⋓ β, but the converse fails in general.When Λ is a finitely aligned LCSC, we can always assume that α ∨ β = Γ where Γ is a finite set of minimal common extensions of α and β. 1.2. Inverse semigroups. Definition 1.9. A semigroup S is an inverse semigroup if for every s ∈ S there exists a unique s * ∈ S such that s = ss * s and s * = s * ss * . Equivalently, S is an inverse semigroup if and only if the subsemigroup E(S) := {e ∈ S : e 2 = e} of idempotents of S is commutative [12, Theorem 1.1.3].A monoid is a semigroup with unit. We say that a semigroup S has zero if there exists 0 ∈ S such that 0s = s0 = 0 for every s ∈ S.Definition 1.10. Given a set X, we define the (symmetric) inverse semigroup on X as

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“…Moreover, since E * is singly aligned, then so is E * ⋊ ϕ G by [2,Proposition 3.12(ii)]. Thus, by [5,Theorem 3.2],…”

Section: Zappa-szép Products Of Lcsc Categoriesmentioning

confidence: 92%

“…Notice that, if Λ is singly aligned, then every filter enjoy condition ( * ) (see e.g. [5,Proposition 3.5]).…”

Section: Filters On Lcscmentioning

confidence: 99%

The tight groupoid of the inverse semigroups of left cancellative small categories

Ortega¹,

Pardo²

2020

Trans. Amer. Math. Soc.

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“…Spielberg observed that all the classes of C * -algebras mentioned above can be viewed as special cases of a general, unifying construction of C * -algebras generated by left regular representations of left cancellative small categories [68,69]. This is a very general construction, as it contains, up to Morita equivalence, all inverse semigroup C * -algebras (see [21]). These C * -algebras come with a distinguished quotient which is called the boundary quotient.…”

Section: Introductionmentioning

confidence: 99%

Left regular representations of Garside categories I. C^*-algebras and groupoids

Li¹

2022

Glasgow Math. J.

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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*-algebras. As another application, we give a complete analysis of the ideal structures of C*-algebras generated by left regular representations of Artin–Tits monoids.

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Broken Möbius categories of $$Q_{3}$$-type and their split inverse semigroups

Schwab

2024

Semigroup Forum

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On zigzag maps and the path category of an inverse semigroup

Cited by 3 publications

References 11 publications

The tight groupoid of the inverse semigroups of left cancellative small categories

The tight groupoid of the inverse semigroups of left cancellative small categories

Left regular representations of Garside categories I. C^*-algebras and groupoids

Broken Möbius categories of $$Q_{3}$$-type and their split inverse semigroups

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On zigzag maps and the path category of an inverse semigroup

Cited by 3 publications

References 11 publications

The tight groupoid of the inverse semigroups of left cancellative small categories

The tight groupoid of the inverse semigroups of left cancellative small categories

Left regular representations of Garside categories I. C*-algebras and groupoids

Broken Möbius categories of $$Q_{3}$$-type and their split inverse semigroups

Contact Info

Product

Resources

About

Left regular representations of Garside categories I. C^*-algebras and groupoids