Amenability versus non amenability: An introduction to von Neumann algebras

Abstract

The theory of von Neumann algebras was initiated by Murray and von Neumann and has deep connections to several areas of mathematics, in particular group theory and ergodic theory. Amenable von Neumann algebras were completely classified by Connes and Haagerup, while numerous classification theorems in the non amenable case were obtained within Popa’s deformation/rigidity theory. This survey article provides an introduction to von Neumann algebras, written for non specialists and with the dichotomy between amenability and non amenability as our guide.