What Operator algebras is a major research subject in pure mathematics. A type of operator algebras called C*-algebras can loosely be divided into two classes, the 'infinite' and the 'finite'. In each class, the so-called 'simple' C*-algebras, which are suitably well-behaved, can be classified. However, the classification results in the finite and the infinite cases, use very different methods. In the last year, I have given a completely new approach to the classification in the infinite case, which has had applications in the study of finite C*-algebras. The aim of this project is to exploit this as well as other recent advances to build a bridge between the infinite and the finite C*-algebras. Why Researchers that study C*-algebras often only consider the finite or the infinite case. My hope is that the theoretical tools that I develop can be used to explain deeply puzzling phenomena, and reveal hidden structure that could not be detected previously. How The research area of operator algebras is purely theoretical and thus does not require any experimental data or similar. However, University of Glasgow, where the project is being carried out, has some of the world leading experts on the subject. I hope to achieve the expected outcome by tight collaboration with these experts, as well as by staying up to date with the newest methods. SSR The research is in pure mathematics with no real life applications in mind.