1
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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 $C_c(\mathbb R)$. $\endgroup$ – Aweygan Mar 4 at 2:35
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – CL. Mar 4 at 8:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.