Graph C⁎-algebras, branching systems and the Perron–Frobenius operator

In this paper we show how to produce a large number of representations of a graph C⁎-algebra in the space of the bounded linear operators in L2(X,μ). These representations are very concrete and, in the case of graphs that satisfy condition (L), we use our techniques to realize the associated graph C⁎-algebra as a subalgebra of the bounded operators in L2(R). We also show how to describe the Perron–Frobenius operator of ergodic theory in terms of the representations we associate to a graph.