Introduction

In 1929, John von Neumann introduced rings of Operators, renamed von Neumann al-

gebras, by proving the fundamental theorem of the subject—the double commutation theo-

rem. He and his collaborator F. J. Murray laid down the foundation for the theory of

Operator algebras in a series of papers [69, 70, 71, 83] during the period from 1936 through

1943. Although there was no immediate follow up, progress in this new field of mathemat-

ics, a steady development and continuous evolution of the theory covered the entire history

after World War II.

Roughly speaking, the theory of Operator algebras, in particular von Neumann algebras,

has two major aspects: the algebraic aspect on one hand and the analytic aspect on the other.

The fundamental question in the theory is how to describe von Neumann algebras on a (sep-

arable) Hubert space. There are several ways to Interpret the question. There have been

and will be several stages and Steps of our understanding of the question. For example, in

the past there was a period when intrinsic characterizations of a von Neumann algebra were

very much emphasized, which brought about Kaplansky's AW*-algebras, Sakai's predual

characterization and the Kadison-Pedersen monotone completeness characterizations. Of

course, what we really want is to describe the structure of algebras in a constructive way,

and in an easy way if possible. The reduction theory brought these questions down to fac-

tors in principle. Thus, the first question is naturally concerned with the number of non-

isomorphic factors. Almost forty years after the first paper of von Neumann, R. Powers and

D. McDuff showed the existence of continuously many nonisomorphic separable factors of

type III and of type II, respectively, thus ending the war against cardinals.

Since our objects are infinite dimensional, the analytic aspect of the subject must be

explored fully in order to answer algebraic questions. Most of the literature in the 50's and

60's is devoted to the analytic side of Operator algebras. This general area is called noncom-

mutative Integration. Before the Tomita-Takesaki theory, noncommutative Integration meant

the theory of traces. But the Tomita-Takesaki theory allows us to include all analytic as-

pects of the theory to noncommutative Integration because we do not have to worry about

the absence of a trace any more. In fact, we will see that a von Neumann algebra of type

III has more structure than semifinite ones.

The Tomita-Takesaki theory and Operator algebraic approach to quantum physics

brought a new item into the subject—automorphisms. The crossed products of Operator al-

gebras was studied by the Japanese school in the late 50's and early 60's, although its origin

1

http://dx.doi.org/10.1090/cbms/051/01