= Cyclic representation of a C-star algebra
{title2=$\overline{\pi(A)\xi}=H$}
A representation of a <C-star algebra> on a <Hilbert space> is a star-homomorphism $\pi:A\to\mathcal B(H)$. It is cyclic if some <vector> $\xi$ has dense orbit span as displayed; the orbit $\pi(A)\xi$ is itself a <linear subspace>. For a unital representation, a unit cyclic <vector> defines a <state on a C-star algebra> by $a\mapsto\langle\pi(a)\xi,\xi\rangle$. Conversely, the <Gelfand–Naimark–Segal construction> realizes every state this way. Cyclicity means density under the action of the whole algebra, not necessarily under powers of a single chosen element.
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