Finite weight projections in von Neumann algebras (Q1174484)

The set of projectors \(P(M)\) of a two-sided ideal \(M\) in a von Neumann algebra \(R\) is a lattice. This fails if \(M\) is one-sided, but \(P(M)\) is still hereditary. However there are interesting cases of one-sided ideals where the set of projections is nevertheless a lattice. If \(\varphi\) is a faithfull semifinite normal weight on \(R\) then \(M_ \varphi=\hbox{span}\{x\in R^ +:|\varphi(x)|\leq\infty\}\) is a hereditary algebra and the criteria are given for \(P(M_ \varphi)\) to be a lattice. These weights are the ones which are comparable with a combination of traces and normal functionals. A central spectral resolution is introduced and used to analyse 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.