= Solution
An <algebra involution> is a conjugate-linear map $a\mapsto a^*$ satisfying $(ab)^*=b^*a^*$ and $(a^*)^*=a$; on a <unital> algebra it fixes $1$. In the Banach-star convention it is continuous, although the following spectral identity uses only its algebraic properties. An element $a$ is invertible precisely when $a^*$ is: its inverse is $(a^{-1})^*$. Applying this to $a=\lambda1-x$ proves
$$
\boxed{\sigma_A(x^*)=\{\overline\lambda:\lambda\in\sigma_A(x)\}}.
$$
For a nonunital algebra, extend the involution to its <unitization of an algebra> by $(a,\lambda)^*=(a^*,\overline\lambda)$ and use the spectrum in that unitization; the same argument proves the identity.
A <unital> <C-star algebra> is a complex <Banach algebra> with an involution satisfying the <C-star identity> $\|a^*a\|=\|a\|^2$. A <Normal element of a C-star algebra> satisfies $x^*x=xx^*$. The <norm> identity for the nonzero identity implies $\|1\|=1$.
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