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

## TL;DR

This paper extends Exel's crossed product construction to non-unital C*-algebras and demonstrates its application to graph C*-algebras, providing criteria for simplicity and ideal structure analysis.

## Contribution

It introduces a generalized crossed product framework for non-unital C*-algebras and applies it to graph C*-algebras, analyzing their structure.

## Key findings

- Realized graph C*-algebras as crossed products
- Provided simplicity criteria for commutative cases
- Analyzed ideal structure of the crossed products

## Abstract

We consider a family of dynamical systems (A,alpha,L) in which alpha is an endomorphism of a C*-algebra A and L is a transfer operator for \alpha. 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.