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 \mathrm 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, \ldots\} , 0<x<1 , (v) all nonnegative reals, (vi) \{0,1\} . Case (i), (ii), (iii) occurs if and only if M is finite type \mathrm I , \mathrm I_∞ hyperfinite type \mathrm{II}_1 , respectively. Case (iv) contains one and only one isomorphic class for each x , and they are type \mathrm{III} . The examples treated by Powers belong to case (iv). Case (v) contains only one isomorphic class and it is type \mathrm{III} . Thus we have a complete classification of factors M which are tensor products of type \mathrm I factors, r_∞(M) ≠ \{0,1\} . Case (vi) contains \mathrm {I}_∞ \otimes hyperfinite \mathrm{II}_1 and also nondenumerably many type \mathrm{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).