Given a real algebraic manifold of the free complex sphere,

and an integer [math]K\in\mathbb N[/math], construct a universal [math]K\times K[/math] model for [math]C(X)[/math],

with [math]T_K[/math] being the space of all [math]K\times K[/math] models for [math]C(X)[/math].

Given [math]X\subset S^{N-1}_{\mathbb C,+}[/math] and [math]K\in\mathbb N[/math] as above, consider the submanifold [math]X^{(K)}\subset X[/math] obtained by factorizing the universal [math]K\times K[/math] model:

Prove that at [math]K=1[/math] we obtain in this way the classical version of [math]X[/math],

and that at [math]K\geq2[/math], assuming that [math]X[/math] is a compact quantum group, [math]X=G\subset U_n^+[/math] with [math]N=n^2[/math], the space [math]X^{(K)}[/math] is not necessarily a compact quantum group.

Work out the details for the fact that the stationarity of a model

implies its faithfulness.

Find an example of an inner faithful model

which is not faithful, not coming from a classical group, or a group dual.

Extract from the general theory developed above a concise proof for the fact that the Pauli matrix model

where [math]x\in SU_2[/math], and [math]c_1,c_2,c_3,c_4[/math] are the Pauli matrices, is faithful.