Let $Gr$ denote the category of discrete groups where the morphisms are group homomorphisms, and let Cstar denote the category of $C^∗$-algebras where the morphisms are $∗$-homomorphisms. Show that $C^*\colon G\to C^*(G)$ is a functor from $Gr$ to $C^*$. In other words, show that any group homomorphism $\psi\colon G\to H$ induces a $∗$-homomorphism $C^*(\psi) \colon C^*(G)\to C^*(H)$ in a natural way such that:
$C^*(id_G) = id_{C^* }(G)$,
If $\psi\colon G\to H$ and $\varphi:H\to K$ are group homomorphisms, then $C^∗(\varphi\circ \psi) =C^∗(\varphi)\circ C^∗(\psi).$
I'm not sure if this question has asked before or not but I have done this, is this the correct way to show functor? If not, what is the other way.
- $C^∗$ preserves identities:
For any group $G$, the identity morphism $id_G: G \to G$ is a group homomorphism. We know, $C^∗(G)$ is the $C^*$-algebra associated with the group $G$. The identity morphism $id_G\colon G \to G$ induces a -homomorphism $C^∗(id_G)\colon C^∗(G) \to C^∗(G)$ by mapping group elements to the corresponding elements in the $C$-algebra.
This $*$-homomorphism $C^∗(id_G)$ is defined as the identity map on the elements of $C^∗(G)$. In other words, for any element $a$ in $C^∗(G)$, we have $C^∗(id_G)(a) = a$.
Hence, $C^∗(id_G) = id_(C^∗(G))$, and the identity preservation condition is satisfied.
- $^∗$ preserves composition:
Consider two group homomorphisms $\psi\colon G \to H$ and $\varphi \colon H \to K$. Let $a$ be an element in $C^∗(G)$. Then $C^∗(\psi\circ\varphi)(a)$ is the image of $a$ under the $*$-homomorphism $C^∗(\psi\circ\varphi)\colon C^∗(G) \circ C^∗(K)$.
$C^∗(\psi\circ\varphi)(a)$ can be computed by first applying $C^∗(\psi)$ to $a$, which maps $a$ to an element $b$ in $C^∗(H)$, and then applying $C^∗(\varphi)$ to $b$, which maps $b$ to an element $c$ in $C^∗(K)$.
We have $$C^∗(\psi\circ\varphi)(a)= C^∗(\varphi)(C^∗(\psi)(a))= C^∗(\varphi)\circ ^∗(\psi)(a).$$
Therefore, $C^∗$ preserves composition.