Given two [math]C^*[/math]-algebras with traces [math]A,B[/math], prove that these algebras are independent inside [math]A\otimes B[/math], and free inside [math]A*B[/math].

Given two discrete groups [math]\Gamma,\Lambda[/math], prove that the algebras [math]C^*(\Gamma),C^*(\Lambda)[/math] are independent inside [math]C^*(\Gamma\times\Lambda)[/math], and free inside [math]C^*(\Gamma*\Lambda)[/math].

Prove that the quantum group inclusion

is an isomorphism, by showing that the corresponding tensor categories coincide.

Work out the details of the identification

and of the corresponding isomorphism at the level of diagonal tori.

Work out a theory of left and right projective versions for the compact quantum groups, and prove that

happens, independently of the projective version theory which is used.