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 L 2 ( 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 L 2 ( R ) . We also show how to describe the Perron–Frobenius operator of ergodic theory in terms of the representations we associate to a graph.