= Gelfand–Naimark–Segal construction
{c}
{title2=$H_x=\overline{A/N_x},\quad N_x=\{a:x(a^*a)=0\}$}
= GNS construction
{c}
{synonym}
For a <state on a C-star algebra> $x$ on a unital algebra $A$, put $\langle[a],[b]\rangle=x(b^*a)$ 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 $c^*c\leq\|c\|^21$ gives $x((ca)^*(ca))\leq\|c\|^2x(a^*a)$, so $N_x$ is a <left ideal> and $[a]\mapsto[ca]$ is well-defined and bounded by $\|c\|$. <Hilbert space completion>[Completion] gives a <Hilbert space> $H_x$ and a star-representation $\pi_x(c)[a]=[ca]$. The adjoint identity follows from $x(b^*ca)=\langle[a],\pi_x(c^*)[b]\rangle$. The unit <vector> $\xi_x=[1]$ is cyclic and recovers the state as $x(a)=\langle\pi_x(a)\xi_x,\xi_x\rangle$.
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