# Contents

## Definition

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

A vector $x \in \mathcal{H}$ is a cyclic vector if $\mathcal{M}x$ is dense in $\mathcal{H}$.

## Properties

The notions of cyclic vector is dual to that of separating 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 cyclic vector appear as vacuum states . See Reeh-Schlieder theorem.

Revised on November 30, 2010 11:42:37 by Urs Schreiber (131.211.232.96)