The Murray — Von Neumann Classification of Factors

Abstract

The notion of unitary equivalence, while being most natural, has the disadvantage of not being additive in the following sense: if e₁, e₂, f₁ and f₂ are Projections such that e_i is unitarily equivalent to f_i, for i = 1,2, and if e₁ ⊥ e₂ and f₁ ⊥ f₂, it is not necessarily true that e₁ + e₂ is unitarily equivalent to f₁ + f₂. This problem disappears if, more generally, one considers two projections as being equivalent if their ranges are the initial and final spaces of a partial isometry. This equivalence, when all the operators concerned -- the projections as well as the partial isometry -- are required to come from a given factor M, is the subject of Section 1.1, where the crucial result is that, with respect to a natural order, the set of equivalence classes of the projections in a factor is totally ordered. The next section examines finite projections -- those not equivalent to proper subprojections; the main result being that finiteness is preserved under taking finite suprema. The final section, via a quantitative analysis of the order relation discussed earlier, effects a primary classification of factors into three types. The principal tool used is called a ‘relative dimension function’ by Murray and von Neumann.