Why is the Gelfand transform injective?

Question

There is a theorem that proves that if $A$ is a commutative C*-Algebra, the Gelfand map is an isometric *-isomorphism of $A$ onto $\hat{A}$ i.e. the spectrum of $A$. (Theorem 1.1 in Averson's "An Invitation to C*Algebras")

It's clear why it's a *-homomorphism. It's clear why it's onto and it's clear why it's isometric. But I don't figure out why it's an isomorphism i.e. why it's injective.

Can you give me a little suggestion? Thank you.

Answer 1

Hint as rhetorical question:

If it's isometric, and $\hat{x} = 0$, what does that tell you about $\lVert x\rVert$?

Answer 2

Injectivity of the Gelfand transform is equivalent to the assertion that characters separate points. This can be verified by using that in the commutative case characters are precisely the pure states and so they have to separate points since the states do.