Cyclic representation of a C-star algebra ($\overline{\pi (A)\xi }=H$)

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 $a\mapsto ⟨\pi (a)\xi ,\xi ⟩$. 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.