= Vector state on a C-star algebra
{title2=$x_\eta(a)=\langle a\eta,\eta\rangle,\quad\|\eta\|=1$}
For a unital represented <C-star algebra> $A\subseteq\mathcal B(H)$, the displayed functional is a state. It is linear under the first-argument-linear inner-product convention, has $x_\eta(1)=1$, and is positive because $x_\eta(a^*a)=\|a\eta\|^2\geq0$. The <Cauchy-Schwarz inequality> gives $|x_\eta(a)|\leq\|a\|$, while evaluation at the unit gives equality of its <norm> with one. Its GNS null <left ideal> consists exactly of the operators annihilating $\eta$.
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