# Publications of the Research Institute for Mathematical Sciences

Volume 4, Issue 1, 1968, pp. 51–130
DOI: 10.2977/prims/1195195263

Published online: 1968-04-30

A Classification of Factors

Huzihiro Araki[1] and E. J. Woods[2]

(1) Kyoto University, Japan
(2) University of Maryland, College Park, USA

A classification of factors is given. For every factor M we define an algebraic invariant r(M), called the asymptotic ratio set, which is a subset of the nonnegative real numbers. For factors which are tensor products of type I factors, the set  r(M) must be one of the following sets: (i) the empty set. (ii) {0}. (iii) {1}, (iv) a one-parameter family of sets {0, xn; n = 0, ±1, ⋯}, 0<x<1, (v) all nonnegative reals, (vi) {0,1}. Case (i), (ii), (iii) occurs if and only if M is finite type I, I hyperfinite type II1, respectively. Case (iv) contains one and only one isomorphic class for each x, and they are type III. The examples treated by Powers belong to case (iv). Case (v) contains only one isomorphic class and it is type III. Thus we have a complete classification of factors M which are tensor products of type I factors, r(M) ≠ {0,1}. Case (vi) contains I ⊗ hyperfinite II1 and also nondenumerably many type III isomorphic classes.
Using the factors in the cases (ii), (iii), (iv) we define another algebraic invariant ρ(M) which is able to distinguish nondenumerably many classes in case (vi).