Exel's crossed product for non-unital C*-algebras

We consider a family of dynamical systems (A, alpha, L) in which a is an endomorphism of a C*-algebra A and L is a transfer operator for a. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.