nLab cyclic vector

Idea

A vector — hence an element of some vector space $\mathcal{H}$ — is called cyclic with respect to the action/representation of some algebra $\mathcal{A}$ on $\mathcal{H}$ if every element of $\mathcal{H}$ may be obtained by acting on $v$ with some algebra element $A \in \mathcal{A}$, or rather if one may always find a sequence of elements $A_n$ whose action on $v$ converges to the given element.

Definition

Definition

Let

1. $\mathcal{A}$ be a C*-algebra;

2. $\mathcal{H}$ a Hilbert space;

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

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

$$im\left( \rho(-)(v) \right) \overset{\text{dense}}{\hookrightarrow} \mathcal{H} \,.$$

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

• In algebraic quantum field theory the states corresponding to cyclic vectors appear as vacuum states. See Reeh-Schlieder theorem.

• GNS construction

References

• eom, Cyclic vector