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 we define an algebraic invariant , called the asymptotic ratio set, which is a subset of the nonnegative real numbers. For factors which are tensor products of type factors, the set  must be one of the following sets: (i) the empty set. (ii) . (iii) , (iv) a one-parameter family of sets , , (v) all nonnegative reals, (vi) . Case (i), (ii), (iii) occurs if and only if is finite type , hyperfinite type , respectively. Case (iv) contains one and only one isomorphic class for each , and they are type . The examples treated by Powers belong to case (iv). Case (v) contains only one isomorphic class and it is type . Thus we have a complete classification of factors which are tensor products of type factors, . Case (vi) contains  hyperfinite  and also nondenumerably many type isomorphic classes.

Using the factors in the cases (ii), (iii), (iv) we define another algebraic invariant 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