= Solution
Let $h=h^*$. The exponential $u_t=e^{ith}$ is defined by its norm-convergent power series. Continuity and conjugate linearity of the <involution> give $u_t^*=e^{-ith}$, and multiplication of the commuting exponential series gives $u_t^*u_t=u_tu_t^*=1$. Thus $u_t$ is a <Unitary element of a C-star algebra> and, by the <C-star identity>, $\|u_t\|=1$.
A <character of an algebra> is automatically continuous with $|\varphi(a)|\le\|a\|$, as proved in Question 5. Passing it through the exponential series gives
$$
|e^{it\varphi(h)}|=|\varphi(u_t)|\le1\qquad(t\in\mathbb R).
$$
The left side is $e^{-t\Im\varphi(h)}$. Considering both signs of $t$ forces $\Im\varphi(h)=0$. So characters take real values on <self-adjoint C-star elements>.
Now write $x=h+ik$ with
$$
h=\frac{x+x^*}{2},\qquad k=\frac{x-x^*}{2i},
$$
both <self-adjoint C-star elements>. Then
$$
\boxed{\varphi(x^*)=\varphi(h)-i\varphi(k)=\overline{\varphi(h)+i\varphi(k)}=\overline{\varphi(x)}.}
$$
This proves that <characters of a C-star algebra respect the involution>. The argument itself does not require commutativity, although a noncommutative algebra need not have any characters.
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