Hi! Let G,H be discrete groups and f:G->H be any homomorphism of these groups. I have three questions about it:
1) how to prove the functoriality of the construction of universal C*-algebra of discrete group (the existence of induced homomorphism C*(G)->C*(H))?
2) how to prove that the construction of reduced C*-algebra of discrete group is not functorial (I am especially interested in counterexamples) and in which case (I mean conditions for group homomorphism) it'll be functorial?
3) let us consider the case when G = Z (integers) and H = Z/ nZ. How to describe the kernel and the image of the induced homomorphism of group C* algebras?