This paper deals with
operator-algebraic aspects of the theory of infinite, locally finite directed graphs. A
(complex-valued) function on the set of edges of a directed graph whose sum over the
edges pointing out of each vertex equals the sum over the edges pointing in is called a
flow. Of particular interest here is the projection of the Hilbert space of
square-summable functions on the edges to the closed subspace consisting of the
square-summable flows. The flow space projection can be identified in a meaningful
and interesting way whenever a group G acts properly on the graph, with the latter
finite modulo the action of G and connected. In general, a choice of vertex and edge
orbit representatives gives a realization of the flow space projection in an
algebra of matrices over the von Neumann algebra of G. Suppressing the
dependence on the choice of orbit representatives yields a class in K0 of this von
Neumann algebra. This K0-class is the sum of the classes arising from the
stabilizers of a representative set of edges minus a corresponding sum for vertices.
Furthermore, if G is non-amenable, all of the foregoing takes place within
the reduced C∗-algebra of G rather than just in the group von Neumann
algebra.
