nLab
cyclic vector

Definition

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

A vector $x\in \mathscr{H}$ is a cyclic vector if $ℳx$ is dense in $\mathscr{H}$.

Properties

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

Applications

In the context of AQFT cyclic vector appear as vacuum states . See Reeh-Schlieder theorem.