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

TL;DR

This paper constructs concrete representations of graph C*-algebras on L^2 spaces and relates them to Perron-Frobenius operators, providing new insights into their structure and operator-theoretic properties.

Contribution

It introduces a method to realize graph C*-algebras as subalgebras of bounded operators on L^2 spaces and connects these to Perron-Frobenius operators.

Findings

Representations of graph C*-algebras in L^2 spaces are explicitly constructed.

Graph C*-algebras satisfying condition (K) are realized as subalgebras of bounded operators on R.

Perron-Frobenius operators are described in L^1 spaces using these representations.

Abstract

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 $L^2(X,\mu)$. These representations are very concrete and, in the case of graphs that satisfy condition (K), 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 some Perron-Frobenius operators in $L^1(X,\mu)$, in terms of the representations we associate to a graph.