Equivalent groupoids have Morita equivalent Steinberg algebras

By L. O. Clark, Aidan Sims. 2013

80 citations recorded by the source corpus.

Let G and H be ample groupoids and let R be a commutative unital ring. We show that if G and H are equivalent in the sense of Muhly–Renault–Williams, then the associated Steinberg algebras are Morita equivalent. We deduce that collapsing a “collapsible subgraph” of a directed graph in the sense of Crisp and Gow does not change the Morita-equivalence class of the associated Leavitt path R -algebra, and therefore a number of graphical constructions which yield Morita equivalent C⁎-algebras also yield Morita equivalent Leavitt path algebras.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f7859fd9053155804f2f626c3b23f63c32bcf16c

How researchers see this paper

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In Algebra and Number Theory

1. Roozbeh Hazrat
2. Aidan Sims
3. Ben Steinberg
4. Huanhuan Li
5. P. Ara
6. Efren Ruiz
7. Toke Meier Carlsen
8. G. Abrams

In Geometry and Topology

1. Aidan Sims
2. Efren Ruiz
3. Roozbeh Hazrat
4. Toke Meier Carlsen
5. Huanhuan Li
6. Ben Steinberg
7. P. Ara
8. Mark V. Lawson

In Mathematical Physics

1. Aidan Sims
2. Roozbeh Hazrat
3. Efren Ruiz
4. Huanhuan Li
5. P. Ara
6. Ben Steinberg
7. Toke Meier Carlsen
8. R. Exel

Citations in the corpus

• Graded Steinberg algebras and their representations (2017, methodology): “We recall from [Clark and Sims 2015, Example 2.1] the construction of the groupoid G E from an arbitrary graph E , which was introduced in [Kumjian et al. 1997] for row-finite graphs and generalised to arbitrary graphs in [Paterson 2002].”
• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras (2016, methodology): “For an ample groupoid G, we write C(G) for the (full) C-algebra of G (see for example [24] or [23]) and, for a commutative ring R with 1, we write AR(G) for the Steinberg algebra of G over R (see [29] and [13]).”
• Simplicity, primitivity and semiprimitivity of etale groupoid algebras with applications to inverse semigroup algebras (2014, methodology): “Let 0 6= ϕ∈ kG and let g∈ G(1) with ϕ(g) 6= 0 and x= d(g) ∈ X. Then ϕ·Indx(Vx) 6= 0 by Lemma 4.3 and hence ϕ·V 6= 0. This proves that V is faithful and hence kG is semiprimitive. The groupoid used in [7] to realize Leavitt path algebras has the property that each isotropy group is either trivial or ”