nLab
cyclic vector

Context

Functional analysis

AQFT

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

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

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

Revised on December 26, 2017 11:47:15 by Daniel M? (2602:306:3668:9e50:fc11:9371:1c95:b14)