= Characters of a C-star algebra respect the involution
{title2=$\phi(x^*)=\overline{\phi(x)}$}
For a self-adjoint element $h$, the exponentials $\exp(ith)$ are unitary. A continuous <algebra character> therefore satisfies $|\exp(it\phi(h))|\leq1$ for every real $t$, forcing $\phi(h)$ 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.
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