Determinant Theory in Finite Factors

Murray and J. v. Neumann classified factors (i.e., central rings of operators) by means of a relative dimension function. They studied, extensively, those factors for which this dimension function has a finite range (finite factors) and showed that these factors (and these factors alone) admit a trace function' with the standard algebraic properties. In an attempt to establish, what seems to us to be a further important algebraic property of the trace, viz., the trace of a generalized nilpotent operator is zero, and, more generally, the trace of an operator lies in the convex hull of its spectrum, we were led to the introduction of a determinant theory for finite factors. This paper will be concerned, principally, with the development of this theory. We might note that it is a simple algebraic matter to prove that the trace T(N) of a proper nilpotent N is zero. In fact, if N' = 0 and E is the projection on the closure of the range of N, then EN = N so that (NE)'-' = N'-'E = 0. Then T(N) = T(EN) = T(NE) = 0, by induction on n. That the normalized trace lies in the convex hull of the spectrum of a finite-dimensional matrix follows at once by bringing the matrix to super-diagonal form, whereby the normalized trace appears as the center of gravity of the spectrum. This fact together with the theory developed in R.O. IV, Chapter IV, yields the same result for operators in an approximately finite factor. Furthermore, it is an immediate consequence of the spectral theorem that the trace of a normal operator in an arbitrary finite factor lies in the convex hull of its spectrum. None of these easily proved facts enabled us to conclude the result for arbitrary non-normal operators in nonapproximately finite factors. However, the general result was established as a byproduct of the determinant theory. In ?2 we define the determinant on regular operators in a factor of type II1,, and establish the properties of this determinant. The proof that the trace lies in the convex hull of the spectrum is given in ?3 as an application of the results of ?2. The uniqueness of the determinant is established in ?4 by means of an algebraic characterization. The final section, ?5, begins with a discussion of the normalization which has taken place in the definition of the determinant. A