Let $G$ and $H$ be discrete groups and $f:G \rightarrow H$ 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 $C^*(G) \to C^*(H)$)?

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 $G = \mathbb{Z}$ (integers) and $H = \mathbb{Z}/ n\mathbb{Z}$. How to describe the kernel and the image of the induced homomorphism of group $C^*$-algebras?

`$C_r^*({\mathbb F}_2)$`

is simple, hence has no non-trivial closed ideals, even though there is an obvious homomorphism from`${\mathbb F}_2$`

onto`${\mathbb Z}^2$`

. Here`${\mathbb F}_2$`

and`${\mathbb Z}_2$`

are, respectively, the free group and the free abelian group on two generators. – Yemon Choi Jun 17 '10 at 14:00`$C^*({\mathbb Z})$`

is. – Yemon Choi Jun 17 '10 at 14:02