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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.”
##### MSC:
 46L30 States of selfadjoint operator algebras
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