nLab separating vector

Definition

Let $\mathcal{M}$ be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$.

A vector $x \in \mathcal{H}$ is a separating vector if $M(x) = 0$ implies $M = 0$ for all $M \in \mathcal{M}$.

Properties

The notions of separating vector is dual to that of cyclic vector with respect to the commutant $\mathcal{M}'$, that is a vector is cyclic for $\mathcal{M}$ iff it is separating for $\mathcal{M}'$.

Applications

In the context of AQFT separating vectors appear as vacuum states . See Reeh-Schlieder theorem.