nLab
cyclic vector

Context

Functional analysis

AQFT

Contents

Idea

A vector, hence an element of some vector space β„‹\mathcal{H} is called cyclic with respect to the action/representation of some algebra π’œ\mathcal{A} on β„‹\mathcal{H} if ever element of β„‹\mathcal{H} may be obtained by acting on vv with some algebra element Aβˆˆπ’œA \in \mathcal{A}, or rather if one may always find a sequence of elements A nA_n whose action on vv converges to the given element.

Definition

Definition

Let

  1. π’œ\mathcal{A} be a C*-algebra;

  2. β„‹\mathcal{H} a Hilbert space;

  3. Ο€:π’œβŸΆβ„¬(β„‹)\pi \;\colon\; \mathcal{A} \longrightarrow \mathcal{B}(\mathcal{H}) a C*-representation of π’œ\mathcal{A} on β„‹\mathcal{H}.

Then a vector vβˆˆβ„‹v \in \mathcal{H} is called a cyclic vector if the image of vv under π’œ\mathcal{A} acting via ρ\rho is a dense subspace of β„‹\mathcal{H}:

im(ρ(βˆ’)(v))β†ͺdenseβ„‹. im\left( \rho(-)(v) \right) \overset{\text{dense}}{\hookrightarrow} \mathcal{H} \,.

Properties

The notions of cyclic vector is dual to that of separating vector with respect to the commutant β„³β€²\mathcal{M}', that is a vector is cyclic for β„³\mathcal{M} iff it is separating for β„³β€²\mathcal{M}'.

Applications

References

Revised on December 3, 2017 14:38:35 by Urs Schreiber (178.6.238.237)