# Graded $C^*$ algebra homomorphism

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 $$\Bbb C = \Bbb R \oplus i \Bbb R$$?

• Not important, but the notation $C_0(\mathbb R)$ usually denotes continuous functions $f:\mathbb R\to\mathbb C$ vanishing at infinity. The standard notation for continuous compactly supported functions is $C_c(\mathbb R)$. – Aweygan Mar 4 at 2:35

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.
• Thanks a lot, if you have time hope you do not mind looking at my new posts, I am quite lost with some definitions of $C^*$ algebras. – CL. Mar 4 at 8:14