Reconstruction: Compact Topological Graphs & Hilbert bimodules over commutative C∗-algebras

We prove that the graph Hilbert bimodule (C∗-correspondence) associated with a compact topological graph can be recovered from three C∗-algebraic data: 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 the work of Davidson–Katsoulis and Davidson–Roydor on local conjugacy of topological graphs. Specifically, we give a direct proof that a compact topological graph can be recovered from its graph module up to local conjugacy, a result to be compared with the strictly stronger and more analytical approach of Davidson–Roydor. From Davidson–Katsoulis Example 3.18 we have an example of nonisomorphic locally conjugate compact topological graphs with isomorphic graph modules; we give another, more concrete, example of this fact. Also, for compact topological graphs with totally disconnected vertex space, we provide a proof that the notions of local conjugacy, Hilbert bimodule isomorphism, isomorphism of C∗-algebraic triples, and isomorphism all coincide, generalizing earlier results on finite directed graphs. We finish by showing that with similar techniques as those we used in compact topological graphs, we can also recover any Hilbert bimodule over a commutative C∗-algebra A whose left action is nondegenerate and by compacts from the C∗-algebraic data: the Toeplitz algebra of the Hilbert bimodule, its gauge action and the coefficient algebra A