The Radon-Nikodym theorem for weights on semifinite JBW-algebras
The purpose of the paper is to obtain a version of the Radon-Nikodym theorem in JBW-algebras - Jordan analogues of von Neumann algebras. The main result can be formulated as follows. 3.6. Corollary. Let A be a JBW-algebra with a faithful normal semifinite trace \(\tau\). if \(\phi\) is a normal semifinite weight on A then there is a unique positive spectral family \((e_{\lambda})\) in A such that \[ \phi (a)=\lim_{\epsilon \to 0}\tau (a\int^{+\infty}_{0}\lambda (1+\epsilon \lambda)^{-1}de_{\lambda}) \] for all \(a\in A^+\). Conversely, every positive spectral family \((e_{\lambda})\) in A defines a normal semifinite weight \(\phi\) by the above formula. Furthermore, the weight \(\phi\) is bounded if and only if \(\int^{+\infty}_{0}\lambda d\tau (e_{\lambda})<+\infty;\) \(\phi\) is a trace if and only if all idempotents \(e_{\lambda}\) are central; \(\phi\) is dominated by \(\tau\) if and only if \(e_{\mu}=1\) for some \(\mu >0\).
scientific article; zbMATH DE number 3905248
scientific article