Restricted Ultraproducts of Finite von Neumann Algebras

Abstract

This chapter discusses the restricted ultra products of finite von Neumann algebras. Considerable progress has been made in the theory of II 1 algebras in the past few years. The restricted ultra product of finite von Neumann algebras is defined to be certain algebra of residue classes of their ultra product, which is a finite von Neumann algebra. The class of finite von Neumann algebras contains all finite dimensional semi simple algebras over the field of complex numbers and all II 1 -factors. Starting especially with an increasing sequence of simple finite dimensional algebras, the restricted ultra product becomes an (infinite dimensional), II 1 -factor.