A Classification of Factors

Abstract

A classification of factors is given. For every factor $M$ we define an algebraic invariant $r_{\infty }(M)$, 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  $r_{\infty }(M)$ must be one of the following sets: (i) the empty set. (ii) $\{0\}$. (iii) $\{1\}$, (iv) a one-parameter family of sets $\{0,x^n;n=0,\pm 1,\dots \}$, $0<x<1$, (v) all nonnegative reals, (vi) $\{0,1\}$. Case (i), (ii), (iii) occurs if and only if $M$ is finite type $\mathrm{I}$, ${\mathrm{I}}_{\infty }$ 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, $r_{\infty }(M)\ne \{0,1\}$. Case (vi) contains ${\mathrm{I}}_{\infty }\otimes$ 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 $\rho (M)$ which is able to distinguish nondenumerably many classes in case (vi).