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Reconstruction of topological graphs and their Hilbert bimodules

Rodrigo Frausino, Abraham C. S. Ng, Aidan Sims

TL;DR

The paper shows that for compact topological graphs, the associated graph bimodule $X(E)$ can be recovered from the Toeplitz algebra triple $(\mathcal{T}C^*(E), \gamma^E, M_E)$, establishing an equivalence between triple isomorphism and bimodule isomorphism. Consequently, for totally disconnected vertex spaces, this reduces to isomorphism of the underlying graphs, and the authors prove a direct link between bimodule isomorphism and local conjugacy of graphs. They also provide an explicit example of nonisomorphic yet locally conjugate graphs with isomorphic graph bimodules and discuss a cohomological obstruction viewpoint, tying the vector-bundle structure of $X(E)$ to permutation-transition data and highlighting limitations of reconstruction in general. The results connect C*-algebraic, graph-theoretic, and topological-bundle perspectives, clarifying when operator-algebraic invariants determine the graph structure and when they only determine local dynamical data.

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.

Reconstruction of topological graphs and their Hilbert bimodules

TL;DR

The paper shows that for compact topological graphs, the associated graph bimodule can be recovered from the Toeplitz algebra triple , establishing an equivalence between triple isomorphism and bimodule isomorphism. Consequently, for totally disconnected vertex spaces, this reduces to isomorphism of the underlying graphs, and the authors prove a direct link between bimodule isomorphism and local conjugacy of graphs. They also provide an explicit example of nonisomorphic yet locally conjugate graphs with isomorphic graph bimodules and discuss a cohomological obstruction viewpoint, tying the vector-bundle structure of to permutation-transition data and highlighting limitations of reconstruction in general. The results connect C*-algebraic, graph-theoretic, and topological-bundle perspectives, clarifying when operator-algebraic invariants determine the graph structure and when they only determine local dynamical data.

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.
Paper Structure (9 sections, 18 theorems, 82 equations)

This paper contains 9 sections, 18 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. KMS-states
  4. Hilbert modules
  5. Topological graphs and modules
  6. Reconstruction of topological--graph bimodules
  7. Local reconstruction of topological graphs
  8. Example
  9. A cohomological obstruction

Key Result

Lemma 2.1

Let $E$ be a compact topological graph. Then the function $v \mapsto |E^1 v|$ is locally constant. For each $v\in E^0$, there exist an open neighborhood $W$ of $v$ and disjoint open s-sections $(Z_e)_{e\in E^1 v}$ such that $s(Z_e)=W$ for all $e\in E^1 v$ and $E^1 W = \bigsqcup_{e\in E^1 v} Z_e$.

Theorems & Definitions (42)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1: hawkins2015applications
  • Theorem 3.2: hawkins2015applications
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 32 more