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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:

[[math]] \pi_K:C(X)\to C(X^{(K)})\subset M_K(C(T_K)) [[/math]]

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

[[math]] X^{(1)}=X_{class} [[/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.