STATEMENT: If A has a multiplicative identity 1, then it is immediate that the equivalence class $ξ$ in the GNS Hilbert space H containing 1 is a cyclic vector for the above representation. If $A$ is non-unital, take an approximate identity $\left\{e_λ\right\}$ for $A$. Since positive linear functionals are bounded, the equivalence classes of the net $\left\{e_λ\right\}$ converges to some vector $ξ$ in $H$, which is a cyclic vector for $π$.
Source: http://en.wikipedia.org/wiki/Gelfand–Naimark–Segal_construction
QUESTION: How do they show that $\xi$ exists. Can someone give a more detailed explanation please. It would be greatly appreciated.
It is perhaps easier to extend the linear functional $\tau$ to a positive linear functional $\tilde{\tau}$ on the unitization $\tilde{A}$. You know get a cyclic representation $(\pi, H, v)$ of $\tilde{A}$. Now restrict $\pi$ to $A$, and we get a representation of $A$. Now we claim that $v$ is a cyclic vector for $\pi\lvert_A$:
Note that $\|\tau\| = \lim_{\lambda} \tau(e_{\lambda}) = \tilde{\tau}(1_{\tilde{A}})$, so $$ \|\pi(e_{\lambda})v - v\|^2 = \tau(e_{\lambda}^2) - 2\tau(e_{\lambda}) + \tilde{\tau}(1_{\tilde{A}}) \leq -\tau(e_{\lambda}) + \|\tau\| \to 0 $$