Find in the literature the complete proof of the GNS theorem, and write down a short account of that, with the main ideas explained.

Find in the literature the complete statement and proof regarding the commutative case, [math]A=L^\infty(X)[/math], and write down a brief account of that.

Try axiomatizing the quadruplets [math](S,T,U,K)[/math] in terms of the associated von Neumann algebras, and report on what you found.

Prove that given a compact quantum group [math]G[/math], in order for having a faithful model [math]C(G)\subset M_K(C(T))[/math], the discrete dual [math]\Gamma=\widehat{G}[/math] must be amenable.

Try to come up with a notion of inner faithfulness for the matrix models [math]C(X)\to M_K(C(T))[/math], in the case where [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is an homogeneous space.

In the context of the matrix truncations, comment on what happens when [math]X_{class}=\emptyset[/math]. Also, comment on the case [math]X^{(\infty)}=X[/math]. And also, comment on the case where [math]X=G[/math] is assumed to be a compact quantum group.

Develop a matrix model theory for the spaces of quantum partial isometries and partial permutations from chapter [math]6[/math].