nLab
separating 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 separating vector if M(x)=0M(x) = 0 implies M=0M = 0 for all MM \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.

Revised on November 30, 2010 11:43:54 by Urs Schreiber (131.211.232.96)