Isomorphisms of tensor algebras of topological graphs

TL;DR

This paper establishes a relationship between algebraic isomorphisms of tensor algebras of topological graphs and the local conjugacy of the graphs, with specific conditions for isometric isomorphisms.

Contribution

It characterizes when tensor algebras of topological graphs are algebraically or completely isometrically isomorphic based on graph conjugacy.

Findings

Algebraic isomorphism implies local conjugacy of graphs.

Under certain conditions, local conjugacy implies isometric isomorphism.

Results connect algebraic properties of tensor algebras with topological graph conjugacy.

Abstract

We show that if two tensor algebras of topological graphs are algebraically isomorphic, then the graphs are locally conjugate. Conversely, if the base space is at most one dimensional and the edge space is compact, then locally conjugate topological graphs yield completely isometrically isomorphic tensor algebras.