Reconstruction of topological graphs and their Hilbert bimodules

We show that the Hilbert bimodule associated to a compact topological graph can be recovered from the C*-algebraic triple consisting of the Toeplitz algebra of the graph, its gauge action and the commutative subalgebra of functions on the vertex space of the graph. We discuss connections with work o...

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Bibliographic Details
Published in:arXiv.org 2023-08
Main Authors: Frausino, Rodrigo, Ng, Abraham C S, Sims, Aidan
Format: Article
Language:English
Subjects:
Algebra
Graphs
Isomorphism
Topology
Online Access:Get full text
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Summary:We show that the Hilbert bimodule associated to a compact topological graph can be recovered from the C*-algebraic triple consisting of the Toeplitz algebra of the graph, its gauge action and the commutative subalgebra of functions on the vertex space of the graph. We discuss connections with work of Davidson-Katsoulis and of Davidson-Roydor on local conjugacy of topological graphs and isomorphism of their tensor algebras. In particular, we give a direct proof that a compact topological graph can be recovered up to local conjugacy from its Hilbert bimodule, present an example of nonisomorphic locally conjugate compact topological graphs with isomorphic Hilbert bimodules. We also give an elementary proof that for compact topological graphs with totally disconnected vertex space the notions of local conjugacy, Hilbert bimodule isomorphism, isomorphism of C*-algebraic triples, and isomorphism all coincide.
ISSN:2331-8422