Every character is a state?

Question

Let $A$ be a unital $C^*$-algebra. Then

$\phi:A\to \mathbb C$ is called a character if it is a non-zero homomorphism.

A character has the following properties:

(1) $\phi(1)=1$.

(2) $\phi(x)\in sp(x):=\{\lambda: \lambda -x \text{ is not invertible}\}$ for all $x\in A$.

(3) $\phi$ is bounded with $\|\phi\|\leq 1$.

Now according to me, (1) and (3) imply:

(4) $\|\phi\|= 1$.

And (2) implies:

(5) $\phi\geq 0$.

$\phi:A\to \mathbb C$ is called a state if it is positive, linear and $\|\phi\|= 1$.

Question: Is it correct that using (4) and (5), every character is a state?

Answer 1

Yes this is correct. Note that more generally, every $*$-homomorphism $\pi: A \to B$ between $C^*$-algebras is positive because $$\pi(a^*a) = \pi(a)^*\pi(a) \ge 0.$$