nLab
cyclic vector

Contents

Context

Functional analysis

AQFT

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

A vector, hence an element of some vector space \mathcal{H} is called cyclic with respect to the action/representation of some algebra 𝒜\mathcal{A} on \mathcal{H} if every element of \mathcal{H} may be obtained by acting on vv with some algebra element A𝒜A \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. 𝒜\mathcal{A} be a C*-algebra;

  2. \mathcal{H} a Hilbert space;

  3. π:𝒜()\pi \;\colon\; \mathcal{A} \longrightarrow \mathcal{B}(\mathcal{H}) a C*-representation of 𝒜\mathcal{A} on \mathcal{H}.

Then a vector vv \in \mathcal{H} is called a cyclic vector if the image of vv under 𝒜\mathcal{A} acting via ρ\rho is a dense subspace of \mathcal{H}:

im(ρ()(v))dense. 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 \mathcal{M}', that is a vector is cyclic for \mathcal{M} iff it is separating for \mathcal{M}'.

Applications

References

Last revised on December 26, 2017 at 11:47:15. See the history of this page for a list of all contributions to it.