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arXiv · 1311.3701

Equivalent groupoids have Morita equivalent Steinberg algebras

Lisa Orloff Clark·Aidan Sims
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Abstract

Let $G$ and $H$ be Hausdorff 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 of locally constant $R$-valued functions with compact support 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.

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Lisa Orloff Clark, Aidan Sims. 2013-11-14. Equivalent groupoids have Morita equivalent Steinberg algebras. https://arxiv.org/abs/1311.3701

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