I - Locally Traceable Operators

Summary

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 A. 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 A. 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 A =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 σ-finite measures, 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 1976] for a very similar purpose to what we have in mind here, and we are pleased to acknowledge our gratitude to him.

To be more formal and more exact about this notion, we consider a separable Hilbert space H with an abelian von Neumann algebra A inside of ℬ(H), the algebra of all bounded operators on H. (We could dispense in part with this separability hypothesis, but it would make life unnecessarily difficult; all the examples and applications we have in mind are separable.) For example, suppose that X is a standard Borel space (see [Arveson 1976] and [Zimmer 1984, Appendix A] for definitions and properties of such spaces). It is a fact that X is isomorphic to either the unit interval [0,1] with the usual σ-field of Borel sets or is a countable set with every subset a Borel set; see [Arveson 1976] for details. Now let µ be a σ-finite measure on X and let H_n be a fixed n-dimensional Hilbert space where n=1, 2 ....,∞.