# Reconstruction of graded groupoids from graded Steinberg algebras

@article{Ara2016ReconstructionOG, title={Reconstruction of graded groupoids from graded Steinberg algebras}, author={Pere Ara and Joan Bosa and Roozbeh Hazrat and Aidan Sims}, journal={arXiv: Rings and Algebras}, year={2016} }

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit.

## 27 Citations

### Diagonal-preserving graded isomorphisms of Steinberg algebras

- MathematicsCommunications in Contemporary Mathematics
- 2018

We study Steinberg algebras constructed from ample Hausdorff groupoids over commutative integral domains with identity. We reconstruct (graded) groupoids from (graded) Steinberg algebras and use this…

### Reconstruction of twisted Steinberg algebras

- Mathematics
- 2021

We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in…

### Diagonal-preserving isomorphisms of étale groupoid algebras

- MathematicsJournal of Algebra
- 2019

### Graded C*-algebras and twisted groupoid C*-algebras

- Mathematics
- 2019

Let $C^*$-algebra that is acted upon by a compact abelian group. We show that if the fixed-point algebra of the action contains a Cartan subalgebra $D$ satisfying an appropriate regularity condition,…

### A note on the core of Steinberg algebras.

- Mathematics
- 2019

In this short note we show that, for an ample Hausdorff groupoid $G$, and the Steinberg algebra $A_R(G)$ with coefficients in the commutative ring $R$, the centraliser of subalgebra $A_R(G^{(0)})$ of…

### Prime étale groupoid algebras with applications to inverse semigroup and Leavitt path algebras

- MathematicsJournal of Pure and Applied Algebra
- 2019

### The polynomial endomorphisms of graph algebras

- Mathematics, Computer ScienceGroups, Geometry, and Dynamics
- 2020

It is shown that the dynamics induced this way on the space of infinite paths (the spectrum of the MASA) is generated by an asynchronous transducer.

### Étale Groupoids and Steinberg Algebras a Concise Introduction

- MathematicsLeavitt Path Algebras and Classical K-Theory
- 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…

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

- MathematicsCanadian Journal of Mathematics
- 2020

Abstract 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…

## References

SHOWING 1-10 OF 36 REFERENCES

### Nonstable K-theory for Graph Algebras

- Mathematics
- 2004

We compute the monoid V(LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property and…

### A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY

- MathematicsJournal of the Australian Mathematical Society
- 2010

Abstract We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and…

### GRAPH INVERSE SEMIGROUPS, GROUPOIDS AND THEIR C -ALGEBRAS

- Mathematics
- 2002

We develop a theory of graph C -algebras using path groupoids and inverse semigroups. Row finiteness is not assumed so that the theory applies to graphs for which there are vertices emitting a…

### Higher-Rank Graph C *-Algebras: An Inverse Semigroup and Groupoid Approach

- Mathematics
- 2004

AbstractWe provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a
uniqueness theorem for the…