Characters of a C-star algebra respect the involution ($\varphi (x^{\ast })=\overline{\varphi (x)}$)

For a self-adjoint element $h$, the exponentials $\exp (ith)$ are unitary. A continuous algebra character therefore satisfies $|\exp (it\varphi (h))|\le 1$ for every real $t$, forcing $\varphi (h)$ to be real. Decomposing an arbitrary element into real and imaginary self-adjoint parts proves the displayed identity. Thus every algebra character is a star-preserving scalar homomorphism; commutativity of the whole algebra is not needed for this fact.