A Classification of Factors

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).