Equivalent groupoids have Morita equivalent Steinberg algebras

@article{Clark2013EquivalentGH,
  title={Equivalent groupoids have Morita equivalent Steinberg algebras},
  author={Lisa Orloff Clark and Aidan Sims},
  journal={Journal of Pure and Applied Algebra},
  year={2013},
  volume={219},
  pages={2062-2075}
}
  • L. O. ClarkA. Sims
  • Published 14 November 2013
  • Mathematics
  • Journal of Pure and Applied Algebra
View PDF on arXiv
67 Citations
Highly Influential Citations
4
Background Citations
19
Methods Citations
8
Results Citations
1

67 Citations

Simple flat Leavitt path algebras are von Neumann regular

Abstract For a unital ring, it is an open question whether flatness of simple modules implies all modules are flat and thus the ring is von Neumann regular. The question was raised by Ramamurthi over…

Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras

We consider the ideal structure of Steinberg algebras over a commutative ring with identity. We focus on Hausdorff groupoids that are strongly effective in the sense that their reductions to closed…

Diagonal-preserving isomorphisms of étale groupoid algebras

Subsets of vertices give morita equivalences of Leavitt path algebras

We show that every subset of vertices of a directed graph E gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an…

On Steinberg algebras of Hausdorff ample groupoids over commutative semirings

Reconstruction of graded groupoids from graded Steinberg algebras

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…

Simplicity of algebras associated to non-Hausdorff groupoids

We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a…

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

  • B. Steinberg
  • Mathematics
    Journal of Pure and Applied Algebra
  • 2019

Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras

We prove that ample groupoids with sigma-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui's notion of Kakutani…

Using the Steinberg algebra model to determine the center of any Leavitt path algebra

Given an arbitrary graph, we describe the center of its Leavitt path algebra over a commutative unital ring. Our proof uses the Steinberg algebra model of the Leavitt path algebra. A key ingredient…
...

References

SHOWING 1-10 OF 33 REFERENCES

Renault's equivalence theorem for reduced groupoid c*-algebras

We use the technology of linking groupoids to show that equivalent groupoids have Morita equivalent reduced C � -algebras. This equivalence is compatible in a natural way in with the Equivalence…

A groupoid generalization of Leavitt path algebras

Let G be a locally compact, Hausdorff groupoid in which s is a local homeomorphism and the unit space is totally disconnected. Assume there is a continuous cocycle c from G into a discrete group…

Simplicity of algebras associated to étale groupoids

We prove that the full C∗-algebra of a second-countable, Hausdorff, étale, amenable groupoid is simple if and only if the groupoid is both topologically principal and minimal. We also show that if G…

Graphs, Groupoids, and Cuntz–Krieger Algebras

We associate to each locally finite directed graphGtwo locally compact groupoidsGandG(★). The unit space ofGis the space of one–sided infinite paths inG, andG(★) is the reduction ofGto the space of…

Characterizations of Morita equivalent inverse semigroups

GRAPH INVERSE SEMIGROUPS, GROUPOIDS AND THEIR C -ALGEBRAS

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…

Strong Morita Equivalence of Inverse Semigroups

We introduce strong Morita equivalence for inverse semigroups. This notion encompasses Mark Lawson's concept of enlargement. Strongly Morita equivalent inverse semigroups have Morita equivalent…

Groupoid models for the C*-algebras of topological higher-rank graphs

We provide groupoid models for Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs. Extending the groupoid models used in the theory of graph algebras and topological dynamical…

Leavitt path algebras with coefficients in a commutative ring

Flow invariants in the classification of Leavitt path algebras