The flow space of a directedG-graph

Abstract

This paper deals with operator-algebraic aspects of the theory of infinite, locally finite directed graphs. A (complex-value d) 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 Hubert space of square-summab le 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 K$ of this von Neumann algebra. This ΛVclass 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. 1. Preliminaries . We will largely follow the notation and terminology of the first chapter of [4] for directed graphs and group actions. A directed graph X consists of a set V of vertices, a set E of edges, and maps /, t: E —• V. The edge y joins the initial vertex i[y) to the terminal vertex t(y). We will assume that (/, t): E —• V x V is injective with range missing the diagonal, i.e. that X has no loops or multiple edges. For a vertex υ , we write staτ(ι ) = i~ι(v)U t~ι(υ), the set of edges incident at , and N(v) = t(i~ι(v)) U i(ΐ~ι(v)), the set of vertices joined to υ by an edge. The cardinality of star(v) is called the degree of we abbreviate deg(v) = |staτ(ι;)|. We will always require X to be locally finite of bounded degree, meaning that sup{deg(w): G V} (which we denote by deg(X)) is finite. (This will ensure that the various Hubert space operators considered below are all bounded.) A path p in X of length n is a sequence v\, y\, ... , υn, yn , vn+ι, where for j = 1, ... , n, the edge yj joins the vertices Vj and Vj+\. We think of p as having a direction of traverse, from v to vn+\, so each edge yy will point either forward or backward along p we set (p, yj) = 1 or -1 depending on whether