Reconstruction of topological graphs and their Hilbert bimodules

TL;DR

This paper demonstrates how to recover the Hilbert bimodule of a compact topological graph from associated C*-algebraic data, establishing connections with graph conjugacy and isomorphism concepts.

Contribution

It provides a method to reconstruct the Hilbert bimodule from C*-algebraic triples and clarifies the relationships between various notions of graph equivalence.

Findings

Hilbert bimodule can be recovered from C*-algebraic data

Nonisomorphic graphs can have isomorphic Hilbert bimodules

For totally disconnected vertex spaces, various graph equivalences coincide

Abstract

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.