Finite weight projections in von Neumann algebras.

The ideal of definition of a faithful semifinite normal weight on a countably decomposable von Neumann algebra is the set generated by all positive elements of finite weight. The set is a hereditary left ideal and therefore contains projections. In this paper the family of weights whose ideals of definition form projection lattices is completely characterized. These weights are the ones that are comparable to a combination of traces and normal functionals. A central spectral resolution is introduced and used to analyze the Radon-Nikodym derivatives of a weight with regard to a trace. Also introduced are two parameters that measure whether the ideal of definition contains two projections of least upper bound 1 and how close the weight is to being a trace respectively.