Condition (K) for inverse semigroups and the ideal structure of their C⁎-algebras

LaLonde, Scott M.; Milan, David; Scott, Jamie

doi:10.1016/j.jalgebra.2018.12.017

Cited by 5 publications

(1 citation statement)

References 22 publications

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“…(this bijection is folklore and can also be essentially found in [19] for tight groupoids of inverse semigroups).…”

Section: Proofmentioning

confidence: 85%

Simplicity of inverse semigroup and étale groupoid algebras

Steinberg¹,

Szakács²

2020

Preprint

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In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal étale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups.The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies.Using Galois descent, we show that simplicity of étale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic 0. We also provide examples of inverse semigroups and étale groupoids with simple algebras outside of a prescribed set of prime characteristics.

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“…(this bijection is folklore and can also be essentially found in [19] for tight groupoids of inverse semigroups).…”

Section: Proofmentioning

confidence: 85%