GCR and CCR Steinberg Algebras

Clark, Lisa Orloff; Steinberg, Benjamin; Wyk, Daniel W. van

doi:10.4153/s0008414x19000415

Cited by 5 publications

(7 citation statements)

References 31 publications

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“…and only if f ′ ≤ f . Thus f is above only finitely many idempotents of D. This proves that (1) implies (2).…”

Section: 1mentioning

confidence: 56%

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Stable finiteness of ample groupoid algebras, traces and applications

Steinberg¹

2022

Preprint

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In this paper we study stable finiteness of ample groupoid algebras with applications to inverse semigroup algebras and Leavitt path algebras, recovering old results and proving some new ones. In addition, we develop a theory of (faithful) traces on ample groupoid algebras, mimicking the C * -algebra theory but taking advantage of the fact that our functions are simple and so do not have integrability issues, even in the non-Hausdorff setting. The theory of traces is closely connected with the theory of invariant means on Boolean inverse semigroups. We include an appendix on stable finiteness of more general semigroup algebras, improving on an earlier result of Munn, which is independent of the rest of the paper.

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“…and only if f ′ ≤ f . Thus f is above only finitely many idempotents of D. This proves that (1) implies (2).…”

Section: 1mentioning

confidence: 56%

“…Later, we shall develop analytic techniques inspired by Passman [17] and C * -algebra theory to address the case of characteristic 0. The main result of this section draws it inspiration from an idea of Munn for monoid algebras [16], together with ideas from [2].…”

Section: Stably Finite Ample Groupoid Inverse Semigroup and Leavitt P...mentioning

confidence: 99%

Stable finiteness of ample groupoid algebras, traces and applications

Steinberg¹

2022

Preprint

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“…We will show that modules for the skew inverse semigroup ring can be identified with sheaves of modules over the sheaf of rings. This will allow the geometric approach to modules initiated in [30], and further studied in [31,14,32], to be applied to skew inverse semigroup rings.…”

Section: Ample Groupoid Convolution Algebras With Coefficients In a Sheaf Of Ringsmentioning

confidence: 99%

Étale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings

Steinberg¹

2020

Can. J. Math.-J. Can. Math.

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Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$ , with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

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“…Then KG| X c is an ideal in KG with KG/KG| X c ∼ = KG| X ; more precisely, restricting an element f ∈ KG to G| X gives a valid element of KG| X (this is not obvious in the non-Hausdorff case) and the kernel of the restriction homomorphism is KG| X c . A proof is given in the discussion following Proposition 5.3 of [33].…”

Section: Proofmentioning

confidence: 99%

Simplicity of inverse semigroup and étale groupoid algebras

Steinberg¹,

Szakács²

2020

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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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GCR and CCR Steinberg Algebras

Cited by 5 publications

References 31 publications

Stable finiteness of ample groupoid algebras, traces and applications

Stable finiteness of ample groupoid algebras, traces and applications

Étale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings

Simplicity of inverse semigroup and étale groupoid algebras

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