# A Classification of Factors

### Huzihiro Araki

Kyoto University, Japan### E. J. Woods

University of Maryland, College Park, USA

## Abstract

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).

## Cite this article

Huzihiro Araki, E. J. Woods, A Classification of Factors. Publ. Res. Inst. Math. Sci. 4 (1968), no. 1, pp. 51–130

DOI 10.2977/PRIMS/1195195263