Singly Generated II_1 Factors

Abstract

In the paper, we study the generator problem of II${}_1$ factors. By defining a new concept related to the number of generators of a von Neumann algebra, we are able to show that a large class of II${}_1$ factors are singly generated, i.e., generated by two self-adjoint elements. In particular, this shows that most of II${}_1$ factors, whose free entropy dimensions are less than or equal to one, are singly generated.