State extensions and a Radon-Nikodým theorem for conditional expectations on von Neumann algebras
Let M be a von Neumann algebra with a faithful normal state \(\omega\) and \(M_ 0\) be a von Neumann subalgebra of M. It is assumed that M acts on a Hilbert space H and the cyclic and separating vector \(\Omega\) determines \(\omega\). Let \(\phi_ 0\) (accordingly \(\omega '_ 0)\) be a faithful normal state on \(M_ 0\) (accordingly \(M'_ 0)\) and set \(\omega '=\omega '_ 0|_{M'}\), \(\omega_ 0=\omega |_{M_ 0}\) and \(\Phi_ 0=(d\phi_ 0/d\omega '_ 0)^{1/2}(d\omega_ 0/d\omega '_ 0)^{- 1/2}\Omega\). The vector-state \({\tilde \phi}{}_ 0(a)=<a\Phi_ 0,\Phi_ 0>\), \(a\in M\), is said to be the canonical extension of \(\phi_ 0\) with respect to \(\omega\) (it does not depend on \(\omega '_ 0)\). It is proved that if \(E_{\omega}: M\to M_ 0\) is the \(\omega\)- conditional expectation, introduced by \textit{L. Accardi} and \textit{C. Cecchini} [J. Funct. Anal. 45, 245-273 (1982; Zbl 0483.46043)], and \(E_{\omega}([{\mathcal D}\phi_ 0,{\mathcal D}\omega_ 0]t)=[{\mathcal D}\phi_ 0,{\mathcal D}\omega_ 0]_ t\) for all \(t\in {\mathbb{R}}\) then \({\tilde \phi}{}_ 0=\phi_ 0\circ E_{\omega}\) (in general \(\phi_ 0\circ E_{\omega}\) is not an extension of \(\phi_ 0)\). Moreover, it is proved the next generalization of Radon-Nikodym's theorem. Let \(\phi\) be a normal state on M and \(\phi_ 0=\phi |_{M_ 0}\) is faithful and \(\phi\leq \lambda {\tilde \phi}_ 0\). Then there exists \(a\in M\) such that \(E_{\omega}(axa^*)=E_{\phi}(x)\) for every \(x\in M\).
von Neumann algebra with a faithful normal state
generalization of Radon-Nikodym's theorem
scientific article; zbMATH DE number 4138553
scientific article