nLab
cyclic vector

Contents

Definition

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

A vector xx \in \mathcal{H} is a cyclic vector if x\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)