The field of operator algebras deals with suitable closed subalgebras of the algebra of bounded linear operators on a Hilbert space. There are two types of operator algebras: C*-algebras and von Neumann algebras. In the 1970s A. Connes obtained a fundamental structural result about amenable von Neumann algebras, leading to a complete classification of these algebras. Since the early 1990s the Elliott classification program has aimed to obtain a corresponding classification result for simple nuclear C*-algebras. In recent years the work of several researchers has provided a classification theorem in the C*-algebras setting, using K-theoretical data. I will describe this classification result by drawing parallels with Connes work, using examples coming from dynamics as a motivation.
