Gelfand–Naimark–Segal construction ($H_x=\overline{A/N_x},N_x=\{a:x(a^{\ast }a)=0\}$)

For a state on a C-star algebra $x$ on a unital algebra $A$, put $⟨[a],[b]⟩=x(b^{\ast }a)$ on $A/N_x$, with the inner product linear in its first argument. The Cauchy–Schwarz inequality for positive C-star functionals makes the null vectors pair to zero with every vector. Moreover $c^{\ast }c\le \Vert c{\Vert }^{2}1$ gives $x((ca{)}^{\ast }(ca))\le \Vert c{\Vert }^{2}x(a^{\ast }a)$, so $N_x$ is a left ideal and $[a]\mapsto [ca]$ is well-defined and bounded by $\Vert c\Vert$. Completion gives a Hilbert space $H_x$ and a star-representation ${\pi }_x(c)[a]=[ca]$. The adjoint identity follows from $x(b^{\ast }ca)=⟨[a],{\pi }_x(c^{\ast })[b]⟩$. The unit vector ${\xi }_x=[1]$ is cyclic and recovers the state as $x(a)=⟨{\pi }_x(a){\xi }_x,{\xi }_x⟩$.

On the C-star algebra of convergent complex sequences, a functional in the displayed coordinate representation is a state on a C-star algebra exactly when all its weights are nonnegative and sum to one. Coordinate projections test positivity of $w_j$ at finite indices. Positive sequences with the first $N$ terms zero and the rest one test $w_{\infty }$ after passage to the limit. The known dual isometry makes the norm the sum of the absolute weights, proving necessity and sufficiency.

If only finitely many weights are positive, the GNS construction identifies the quotient by $\{\alpha :F({\alpha }^{\ast }\alpha )=0\}$ with the values at precisely those supporting indices. Its inner product is the weighted coordinate inner product. Passing to the standard Hilbert norm gives diagonal evaluation operators and cyclic vector whose entries are the square roots of the weights. Arbitrary finite coordinate values and the limiting value can be assigned independently by a convergent sequence, which proves cyclicity. For weights $1/2$ at infinity and $1/4$ at each of the first two indices, the space is ${\mathbb{C}}^3$ and the cyclic vector is $(1/\sqrt{2},1/2,1/2)$.

For the state on a C-star algebra $a\mapsto \mathrm{Tr}(\rho a)$ on $M_d(\mathbb{C})$, the GNS construction has the displayed Hilbert-space dimension. Here $\rho$ is a density matrix. Its null space for the GNS seminorm consists of $b$ with $b{\rho }^{1/2}=0$, because $\mathrm{Tr}(\rho b^{\ast }b)=\Vert b{\rho }^{1/2}{\Vert }_{\mathrm{HS}}^2$. Thus the GNS space is the space of matrices $b{\rho }^{1/2}$ with the Hilbert-Schmidt inner product. If $\rho$ has rank $r$, diagonalization identifies these matrices with $r$ arbitrary columns, giving dimension $dr$. Left multiplication is the representation and ${\rho }^{1/2}$ is its cyclic vector. In particular a pure rank-one matrix state gives the defining $d$-dimensional representation, while a full-rank state gives a $d^2$-dimensional space.

For a vector state on a C-star algebra, $N=\{a:a\eta =0\}$ and $⟨[a],[b]⟩=⟨a\eta ,b\eta ⟩$. 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 $v\mapsto ⟨v,\eta ⟩\zeta$ sends $\eta$ to any chosen $\zeta$. In this case, for the rank-one orthogonal projection $p_{\eta }$, $a\eta =0$ is equivalent to $a{p}_{\eta }=0$, by evaluating at $\eta$ and using $(a{p}_{\eta })v=⟨v,\eta ⟩a\eta$.