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Relatively ”closable” states on \(C^*\)-algebras. (English) Zbl 0575.46050
”Let \({\mathcal C}\) be a \(C^*\)-algebra and \(\phi\) a state on \({\mathcal C}\). Let (\({\mathcal H}_{\phi},\pi_{\phi},\xi_{\phi})\) be the GNS- representation of \({\mathcal C}\) and denote by \(S_{\phi}\) the set of states \(\psi\) on \({\mathcal C}\) implemented by vectors in the cone \([\pi_{\phi}({\mathcal C})'\!_+\xi_{\phi}]\), that is \[ \psi =\omega_{\xi}\circ \pi_{\phi},\quad \omega_{\xi}=(\cdot \xi,\xi) \] for some \(\xi =\xi_{\psi}\) in the closure \([\pi_{\phi}({\mathcal C})'\!_+\xi_ p]\) of \(\pi_{\phi}({\mathcal C})'\!_+\xi_{\phi}.\)
It is demonstrated that a state \(\psi\) lies in \(S_{\phi}\) if and only if there is a positive map E of \({\mathcal C}\) into an Abelian von Neumann algebra \({\mathcal A}\) and positive normal functionals (or states) \({\bar \phi}\), \({\bar \psi}\) on \({\mathcal A}\), so that \({\bar \phi}\) is faithful and \(\phi ={\bar \phi}\circ E,\psi ={\bar \psi}\circ E\). Further it is shown that \(S_{\phi}\) consists of precisely the states \(\psi\) on \({\mathcal C}\) which are ’closable’ with respect to \(\phi\) in the sense that the map \[ {\mathcal H}_{\phi}\ni \pi_{\phi}({\mathcal C})\xi_{\phi}\to \pi_{\psi}({\mathcal C})\xi_{\psi}\in {\mathcal H}_{\psi} \] is well- defined and closable.
The proof of these assertions is based on the ”if” part of the following statement: When \({\mathcal M}\) is a von Neumann algebra with a cyclic and separating vector \(\xi\) and \({\mathcal B}\) is a von Neumann subalgebra of \({\mathcal M}'\), there is a normal, conditional expectation of \({\mathcal M}\vee {\mathcal B}\) onto \({\mathcal B}\) preserving \(\omega_{\xi}\) if and only if \({\mathcal B}\) is Abelian.”
46L30 States of selfadjoint operator algebras
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