Gelfand–Naimark–Segal construction

Remarks

Then $\pi_\rho$ is a $*$-representation of $A$, the vector $\xi_\rho$ is cyclic, that is, $\pi_\rho(A)\xi_\rho$ is dense in $H_\rho$, and $\rho(a) = \ip{\pi_\rho(a)\xi_\rho}{\xi_\rho}$ for all $a \in A$. Any other cyclic representation with a cyclic vector $\xi$ satisfying $\rho(a) = \ip{\pi(a)\xi}{\xi}$ is unitarily equivalent to $\pi_\rho$ by a unitary sending $\xi$ to $\xi_\rho$. The boundedness in clause 3 comes from $\rho(b^*a^*ab) \le \norm{a}^2 \rho(b^*b)$.

Example: $A = C[0, 1]$ and $\rho(f) = \int_0^1 f(x) \, dx$. Then $N_\rho = \set{0}$, since $\int_0^1 \abs{f}^2 \, dx = 0$ forces a continuous $f$ to vanish, and the inner product is $\int_0^1 f \overline{g} \, dx$. So $H_\rho = L^2[0, 1]$, $\pi_\rho(f)$ is multiplication by $f$ and $\xi_\rho$ is the constant function $1$.

Non-example: the functional $\sigma(f) = -f(0)$ on $C[0, 1]$ is linear but not a state, and the construction fails at clause 1. The form $\ip{a}{b} = \sigma(b^*a)$ gives $\ip{1}{1} = -1 < 0$, so it is not an inner product.

For a non-unital C*-algebra the construction works in the same way for positive functionals, with the cyclic vector obtained from an approximate unit.