= GNS representation of a vector state
{c}
{title2=$U[a]=a\eta,\quad U\pi_{x_\eta}(a)U^{-1}=a|_{\overline{A\eta}}$}
For a <vector state on a C-star algebra>, $N=\{a:a\eta=0\}$ and $\langle[a],[b]\rangle=\langle a\eta,b\eta\rangle$. Thus $U_0[a]=a\eta$ is a well-defined <linear isometry> from the quotient pre-Hilbert space onto $A\eta$. It extends to an <isometry> from the completed GNS space onto $K=\overline{A\eta}$: its range is complete and therefore closed, and contains the dense subspace $A\eta$. The space $K$ is invariant under both $A$ and its adjoints. On quotient <vectors>, $U\pi(a)[b]=ab\eta=aU[b]$, proving the displayed unitary equivalence after <continuity>. The cyclic <vector> maps to $\eta$.
If $A=\mathcal B(H)$, then $A\eta=H$, since the rank-one map $v\mapsto\langle v,\eta\rangle\zeta$ sends $\eta$ to any chosen $\zeta$. In this case, for the rank-one <orthogonal projection> $p_\eta$, $a\eta=0$ is equivalent to $ap_\eta=0$, by evaluating at $\eta$ and using $(ap_\eta)v=\langle v,\eta\rangle a\eta$.
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