Locally Traceable Operators

Abstract

Our object in this chapter is to develop the notion of what we call locally traceable operators -- or, more or less equivalently, the notion of locally finite dimensional subspaces relative to an abelian von Neumann algebra α. The underlying idea here is that certain operators, although not of trace class in the usual sense, are of trace class when suitably localized relative to α. The trace, or perhaps better, the local trace of such an operator is not any longer a number, but is rather a measure on a measurable space X associated to the situation with α= L^∞(X). This measure is in general infinite but σ-finite, and it will be finite precisely when the operator in question is of trace class in the usual sense, and then its total mass will be the usual trace of the operator. Heuristically, the local trace, as a measure, will tell us how the total trace — infinite in amount — is distributed over the space X. Once we have the notion of a locally traceable operator, and hence the notion of locally finite dimensional subspaces, one can define then the local index of certain operators. This will be the difference of local dimensions of the kernel and cokernel, and will therefore be, as the difference of two a-finite measure, a σ-finite signed measure on X. One has to be slightly careful about expressions such as ∞ — ∞ that arise, but this is a minor matter and can be avoided easily by restricting consideration to sets of finite measure. These ideas are developed to some extent in Atiyah [At3] for a very similar purpose to what we have in mind here, and we are pleased to acknowledge our gratitude to him.