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

Abstract

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 equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras $L_Z(E_2)$ and $L_Z(E_{2-})$ are not stably *-isomorphic.

12 pages. Minor corrections. This is the version that will be published

References in corpus (3)

• Renault's Equivalence Theorem for Reduced Groupoid C*-algebras
• A Groupoid Approach to Discrete Inverse Semigroup Algebras
• $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$

Cited by in corpus (28)

• Diagonal-preserving gauge-invariant isomorphisms of graph $C^{\ast }$-algebras
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• Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
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• Diagonal-preserving graded isomorphisms of Steinberg algebras
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• Rigidity of ${\ell }^p$ Roe-type algebras
• Orbit equivalence of graphs and isomorphism of graph groupoids
• $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$
• Graded C*-algebras and twisted groupoid C*-algebras
• C*-algebras, groupoids and covers of shift spaces
• Refined moves for structure-preserving isomorphism of graph C*-algebras
• The type semigroup, comparison and almost finiteness for ample groupoids
• Monic representations of finite higher-rank graphs
• A dynamical characterization of diagonal preserving $\ast$-isomorphisms of graph $C^{\ast }$-algebras
• A dichotomy for groupoid C*-algebras
• Cartan subalgebras in C*-algebras. Existence and uniqueness
• On tracial $\mathcal{Z}$-stability of simple non-unital C*-algebras
• Flow equivalence of topological Markov shifts and Ruelle algebras
• One-sided topological conjugacy of topological Markov shifts, continuous full groups and Cuntz--Krieger algebras
• Dynamic characterizations of quasi-isometry, and applications to cohomology
• Separable representations of higher-rank graphs
• The Cuntz splice does not preserve $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$
• Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts
• On one-sided topological conjugacy of topological Markov shifts and gauge actions on Cuntz--Krieger algebras
• Exchange rings and real rank zero C*-algebras associated with finitely separated graphs
• Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras
• Irreducibility and monicity for representations of $k$-graph $C^{\ast }$-algebras