# Publications of the Research Institute for Mathematical Sciences

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**Volume 4, Issue 1, 1968, pp. 51–130**

**DOI: 10.2977/prims/1195195263**

A Classification of Factors

Huzihiro Araki^{[1]}and E. J. Woods

^{[2]}(1) Kyoto University, 606-8502, KYOTO, JAPAN

(2) Department of Physics and Astronomy, University of Maryland, MD 20742, COLLEGE PARK, UNITED STATES

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).

*No keywords available for this article.*

Araki Huzihiro, Woods E.: A Classification of Factors. *Publ. Res. Inst. Math. Sci.* 4 (1968), 51-130. doi: 10.2977/prims/1195195263