Why isn't complex conjugation a *-homomorphism?

Question

By Gelfand-duality with $1$, the set $\{pt\}$ is homeomorphic to $\mathrm{Spec}(C(\{pt\})) \cong \mathrm{Spec}(\mathbb{C})$, the set of nonzero *-homomorphisms $\mathbb{C} \rightarrow \mathbb{C}$ with the pointwise convergence topology.

In particular, there can be only one * -homomorphism $\mathbb{C} \rightarrow \mathbb{C}$. But why can't I take both $id$ and complex conjugation (*) itself?

Look at the definition of $*$-homomorphism. – GEdgar Oct 04 '13 at 00:26 — GEdgar, Oct 04 '13 at 00:26

score 1 · Accepted Answer · answered Oct 07 '13 at 12:25

1

By definition $^*$-homomorphism is a linear map, but complex conjugation is antilinear

answered Oct 07 '13 at 12:25

Norbert

56,803

Why isn't complex conjugation a *-homomorphism?

1 Answers1