Title: Characterizing Graph C*-Correspondences

To any directed graph E=(E^0,E^1,r,s) one can associate a certain
C*-correspondence X_E called the graph correspondence.  It has been observed
that the Cuntz-Pimsner algebra O_{X_E} is isomorphic to the graph algebra C*(E).
We shall obtain a characterization of the graph correspondence associated to a
directed graph E. Namely, every nondegenerate C*-correspondence over a
commutative C*-algebra with discrete spectrum is isomorphic to a graph
correspondence.  We will then describe a functor between the category G of
directed graphs with vertex set V whose morphisms are injective graph morphisms
and the category C_0 of nondegenerate C*-correspondences over c_0(V) whose
morphisms will be morphisms of c_0(V)-correspondences.  We will show that this
functor has some but not all of the properties of being an equivalence of
categories.