Topological Full Groups of Ample Groupoids with Applications to Graph Algebras

Abstract

We study the topological full group of ample groupoids over locally compact spaces. We extend Matui's definition of the topological full group from the compact, to the locally compact case. We provide two general classes of groupoids for which the topological full group, as an abstract group, is a complete isomorphism invariant. Hereby extending Matui's Isomorphism Theorem. As an application, we study graph groupoids and their topological full groups, and obtain sharper results for this class. The machinery developed in this process is used to prove an embedding theorem for ample groupoids, akin to Kirchberg's Embedding Theorem for $C^{\ast }$-algebras. Consequences for graph $C^{\ast }$-algebras and Leavitt path algebras are also spelled out. In particular, we improve on a recent embedding theorem of Brownlowe and Sørensen for Leavitt path algebras.

58 pages. V4: Deleted an obsolete reference. Updated Journal reference. Minor typographical changes. (V3: Rewritten the "Our results" and "Précis" part of the Introduction. Added Remark 6.20 and Definition 7.8.)

References in corpus (6)

• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
• Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
• Cartan subalgebras in C*-algebras of Hausdorff etale groupoids
• Diagonal-preserving graded isomorphisms of Steinberg algebras
• Orbit equivalence of graphs and isomorphism of graph groupoids
• Leavitt $R$-algebras over countable graphs embed into $L_{2,R}$

Cited by in corpus (2)

• Representing topological full groups in Steinberg algebras and C*-algebras
• Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras