Abstract : In the world of von Neumann algebras, factors can be classified into three types. The type III factors are those that do not have a trace. They are related to nonsingular ergodic actions, regular representations of non-unimodular groups and quantum field theory. Some of the key structural properties of this class of factors are still not well understood. In this mini-course, I will give a gentle introduction to the theory of type III factors and to the deepest open problem in the theory : Connes's Bicentralizer Problem (1979).

Keywords : von Neumann algebra; type III factor; Tomita-Takesaki; Bicentralizer problem