Twisted topological graph algebras

We generalize Katsura's graph correspondence associated to a topological graph by incorporating a 1-cocycle on the edge set of the graph and constructing a C*-correspondence called the twisted graph correspondence. We investigate the Cuntz-Pimsner algebra of the twisted graph correspondence by introducing covariant twisted Toeplitz representations of the graph, and showing that the Cuntz-Pimsner algebra of the twisted graph correspondence is generated by a universal covariant twisted Toeplitz representation of the graph. We expand on Katsura's ideas to prove fundamental results about the Cuntz-Pimsner algebra of the twisted graph correspondence. In particular we establish a version of the Cuntz-Krieger Uniqueness Theorem, and study the ideal structure.