Graded $C^*$ algebra homomorphism

Question

I am pretty sure I have some definition wrong. But I do not see where. Here is the context:

---

Consider the $C^*$ algebra of continuously compactly supported functions $\Bbb R$ into $\Bbb C$.

$$C_0(\Bbb R)$$ Then there is a grading giving by even and odd functions.

It is claimed that the map $f \mapsto f(0)$, $$ C_0(\Bbb R) \rightarrow \Bbb C$$

Is a graded $*$-homomoprhism.

---

This does not make sense to me, if $f$ is even function, there is no restriction on $f(0)$. In particular, it would not be a graded morphism to the grading on $\Bbb C = \Bbb R \oplus i \Bbb R$?

Answer 1

The grading on $\mathbb C$ is given by the decomposition $(\mathbb C)^+=\mathbb C$ and $(\mathbb C)^-=\{0\}$. With this grading, the map $f\mapsto f(0)$ is a graded $*$-homomorphism.