= States and finite-support GNS models of convergent sequences
{title2=$F(\alpha)=w_\infty\alpha_\infty+\sum_jw_j\alpha_j$}
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 $\{\alpha:F(\alpha^*\alpha)=0\}$ 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 $(1/\sqrt2,1/2,1/2)$.
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