# Strongly graded groupoids and strongly graded Steinberg algebras

@article{Clark2017StronglyGG, title={Strongly graded groupoids and strongly graded Steinberg algebras}, author={L. O. Clark and R. Hazrat and S. Rigby}, journal={arXiv: Rings and Algebras}, year={2017} }

We study strongly graded groupoids, which are topological groupoids $\mathcal G$ equipped with a continuous, surjective functor $\kappa: \mathcal G \to \Gamma$, to a discrete group $\Gamma$, such that $\kappa^{-1}(\gamma)\kappa^{-1}(\delta) = \kappa^{-1}(\gamma \delta)$, for all $\gamma, \delta \in \Gamma$. We introduce the category of graded $\mathcal G$-sheaves, and prove an analogue of Dade's Theorem: $\mathcal G$ is strongly graded if and only if every graded $\mathcal G$-sheaf is induced… Expand

#### 13 Citations

Crossed product Leavitt path algebras

- Mathematics
- 2020

If $E$ is a directed graph and $K$ is a field, the Leavitt path algebra $L_K(E)$ of $E$ over $K$ is naturally graded by the group of integers $\mathbb Z.$ We formulate properties of the graph $E$… Expand

Strongly graded groupoid and directed graph algebras.

- Mathematics
- 2020

We show the reduced $C^*$-algebra of a graded ample groupoid is a strongly graded $C^*$-algebra if and only if the corresponding Steinberg algebra is a strongly graded ring. We apply this result to… Expand

Equivariant dimensions of graph C*-algebras

- Mathematics
- 2019

We explore the recently introduced local-triviality dimensions by studying gauge actions on graph $C^*$-algebras, as well as the restrictions of the gauge action to finite cyclic subgroups. For… Expand

Gauge freeness for Cuntz-Pimsner algebras

- Mathematics
- 2018

To every $C^*$ correspondence over a $C^*$-algebra one can associate a Cuntz-Pimsner algebra generalizing crossed product constructions, graph $C^*$-algebras, and a host of other classes of operator… Expand

Properties of the gradings on ultragraph algebras via the underlying combinatorics

- Mathematics
- 2021

There are two established gradings on Leavitt path algebras associated with ultragraphs, namely the grading by the integers group and the grading by the free group on the edges. In this paper, we… Expand

R A ] 2 0 A ug 2 02 0 A NOTE ON THE REGULAR IDEALS OF LEAVITT PATH ALGEBRAS

- 2020

We prove algebraic versions of recent results, proved by Brown, Fuller, Pitts, and Reznikoff, regarding regular and gaugeinvariant ideals of graph C*-algebras. More precisely, for Leavitt path… Expand

Realizing ultragraph Leavitt path algebras as Steinberg algebras

- Mathematics
- 2020

In this article, we realize ultragraph Leavitt path algebras as Steinberg algebras. This realization allows us to use the groupoid approach to obtain structural results about these algebras. Using… Expand

Étale Groupoids and Steinberg Algebras a Concise Introduction

- Mathematics
- 2020

We give a concise introduction to (discrete) algebras arising from etale groupoids (aka Steinberg algebras) and describe their close relationship with groupoid \(C^*\)-algebras. Their connection to… Expand

O A ] 3 1 M ay 2 01 8 Gauge freeness for Cuntz-Pimsner algebras

- 2018

To every C correspondence over a C-algebra one can associate a Cuntz-Pimsner algebra generalizing crossed product constructions, graph C-algebras, and a host of other classes of operator algebras.… Expand

A note on the regular ideals of Leavitt path algebras

- Mathematics
- 2020

We prove algebraic versions of recent results, proved by Brown, Fuller, Pitts, and Reznikoff, regarding regular and gauge-invariant ideals of graph C*-algebras. Precisely, for Leavitt path algebras… Expand

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