If $(R,H)$ is a von Neumann algebra, then what is the expectation functional by a vector $x \in H$?

Question

Let $(R,H)$ be a von Neumann algebra. Let $\rho$ be a state on $R$.

In the abstract of this paper the author says

... the expection value by the vector $\sigma_{\psi}(\rho)$ is $\rho$.

Now I was wondering what this meant. My guess is that we have \begin{equation} \rho(r) = \langle r \cdot \sigma_{\psi}(\rho), \sigma_{\psi}(\rho) \rangle_{H}, \end{equation} for all $r \in R$. Is this correct? I have tried looking further into the paper, but I was unable to verify my guess.

Answer 1

Yes, it means exactly what you wrote. The name comes from Quantum Mechanics.