In the algebraic formulation of Quantum Mechanics, how do probability amplitudes naturally arise? In the algebraic formulation of quantum mechanics, consider $\mathcal{B}(\mathcal{H})$ as the set of all bounded operators on $\mathcal{H}$ (with involution, norm, etc.), which form a C*-algebra $C$. States are defined as linear functionals (dual space of $\mathcal{B}(\mathcal{H})$), $\phi: C \mapsto \mathbb{C}$.
My question is from the above construction, how does $\phi$ naturally describe a probability amplitude?
 A: You can certainly define the probability amplitude of a pair of pure states which are normal states with respect to a given algebraic state and this mathematical object has the same properties as in the standard formulation.
When you have an algebraic state on the $C^*$-algebra $A$, that is a positive ($\phi(a^*a)\geq 0$), normalized ($\phi(I)=1$), linear functional $\phi : A \to \mathbb C$, you can represent it in a Hilbert space by means of the GNS construction. 
Up to unitary isomorphisms there is a triple $(\cal H, \pi, \Psi)$, where 
(i) $\cal H$ is a Hilbert space, 
(ii) $\pi: A \to B(\cal H)$ a (continuous) $^*$-representation,
(iii) $\Psi \in \cal H$ a unit vector 
such that 
(a) $\pi(A)\Psi$ is dense in $\cal H$ 
and 
(b) $\phi(a)= \langle \Psi|\pi(a) \Psi\rangle$.
For the sake of semplicity let us henceforth assume that $\phi$ is pure (i.e. an extreme element of the convex set of algebraic states). 
The vectors $\Phi \in \cal H$ represent (up to normalization and a phase) other pure states of the system, the so-called  normal pure state of the system in the folium of $\phi$ 
N.B. If $\dim(\cal H) = \infty$, there are many other algebraic pure states which cannot be represented as vectors in $\cal H$.
These states are of the form $\pi(a)\Psi$ ($a\neq 0$) or limit of sequences $\pi(a_n)\Psi$ of such vectors due to (a) above. 
Sticking to the most elementary case of a normal pure state defined by the vector  $\pi(a)\Psi$. Algebraically it is defined by the functional
$\phi_a(b) = \frac{\langle \pi(a)\Psi | \pi(b) \pi(a) \Psi\rangle}{\langle \pi(a)\Psi |  \pi(a) \Psi\rangle}$ namely taking advantage of the GNS construction (especially (b) above)
$$A \ni b \mapsto \phi_a(b) = \frac{\omega(a^*ba)}{\omega(a^*a)}\tag{1}\:.$$
The probability amplitude of $\Phi_1 = \phi(a_1)\Psi$ and $\Phi_2 = \phi(a_2)\Psi$ is, as usual,
$$\frac{\langle \Phi_1| \Phi_2 \rangle}{||\Phi_1|| ||\Phi_2||} = \frac{\langle \pi(a_1)\Psi| \pi(a_2)\Psi\rangle}{||\Phi_1|| ||\Phi_2||} = \frac{\langle \Psi| \pi(a_1)^*\pi(a_2)\Psi\rangle}{||\Phi_1|| ||\Phi_2||}= \frac{\langle \Psi| \pi(a_1^*a_2)\Psi\rangle}{||\Phi_1|| ||\Phi_2||}
= \frac{\langle \Psi| \pi(a_1^*a_2)\Psi\rangle}{\sqrt{\phi(a_1^*a_1)\phi(a_2^*a_2)}} = \frac{\phi(a_1^*a_2)}{\sqrt{\phi(a_1^*a_1)\phi(a_2^*a_2)}}\:.$$
We conculide that, given an algebraic pure state $\phi$ and considering two normal pure states $\phi_{a_i}$, $a_1,a_2 \in A$ defined by (1), the probability amplitude of $\phi_{a_1}$ and $\phi_{a_2}$ is defined as
$$PA(\phi_{a_1},\phi_{a_2}) := \frac{\phi(a_1^*a_2)}{\sqrt{\phi(a_1^*a_1)\phi(a_2^*a_2)}}\:.$$
