# 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 ${0,x_{n};n=0,±1,…}$, $0<x<1$, (v) all nonnegative reals, (vi) ${0,1}$. Case (i), (ii), (iii) occurs if and only if $M$ is finite type $I$, $I_{∞}$ hyperfinite type $II_{1}$, 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 $II_{1}$ 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