Clarification of the notion "faithful" of representations of $C^*$-algebras

Question

Let $A$ be a $C^*$-algebra and $\pi:A\to B(H)$ a $*$-representation for $A$ ($H$ is a Hilbert space). From lecture I learned that we call "$\pi$ is faithful" if $\pi$ is injective. But why is it not allowed to say that $\rho:A\to \mathbb{C}$ is faithful, if $\rho$ is an injective state of $A$? I guess that it's also forbidden to say "faithful" for injective $*$-homomorphisms $f:A\to B$ on $C^*$-algebras $A$ and $B$.

My question is therefore: What does "faithful" really mean? Where does the notion "faithful" come from?

Answer 1

According to this Wolfram page faithfulness is a property of the pair $(H,\phi)$ where $H$ is a Hilbert space and $\phi:A\rightarrow B(H)$ is an injective $*$-homomorphism.

I think the following example there is helpful.

Answer 2

For a $*$-homomorphism, faithful is the same as one-to-one (I usually avoid injective because it has another meaning in the theory).

It is precisely for states where the difference is significant. A functional can only be one-to-one if its domain is one-dimensional. That is, unless $A=\mathbb C$, any state on any C$^*$-algebra is not one-to-one. On the other hand, many states are faithful and these are particularly important states (among other things, because their corresponding GNS representations are faithful).

As for the origin of the name, as Luiz mentioned, it probably has to do with the fact that a faithful representation gives you an embedding.