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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.
Let
$\mathcal{A}$ be a C*-algebra;
$\mathcal{H}$ a Hilbert space;
$\pi \;\colon\; \mathcal{A} \longrightarrow \mathcal{B}(\mathcal{H})$ 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 $\mathcal{H}$:
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 $\mathcal{M}'$.
In algebraic quantum field theory the states corresponding to cyclic vectors appear as vacuum states. See Reeh-Schlieder theorem.
Last revised on December 26, 2017 at 11:47:15. See the history of this page for a list of all contributions to it.