An elementary approach to C∗-algebras associated to topological graphs

We develop notions of a representation of a topological graph E and of a covariant representation of a topological graph E which do not require the machinery of C∗-correspondences and Cuntz– Pimsner algebras. We show that the C∗-algebra generated by a universal representation of E is isomorphic to the Toeplitz algebra of Katsura’s topological-graph bimodule, and that the C∗-algebra generated by a universal covariant representation of E is isomorphic to Katsura’s topological graph C∗-algebra. We exhibit our results by constructing the isomorphism between the C∗-algebra of a row-finite directed graph E with no sources and the C∗-algebra of the topological graph arising from the shift map acting on the infinite-path space E∞.