= Solution
The <spectrum of an element> is $\sigma_A(a)=\{\lambda\in\mathbb C:a-\lambda1_A\text{ is not invertible in }A\}$. A <Normal element of a C-star algebra> satisfies $a^*a=aa^*$.
For a <Hermitian C-star element> $h$, the <C-star identity> gives $\|h^2\|=\|h\|^2$. For normal $a$, commutation gives
$$
\|a^2\|^2=\|(a^*)^2a^2\|=\|(a^*a)^2\|=\|a^*a\|^2=\|a\|^4.
$$
All powers remain normal, so induction yields $\|a^{2^k}\|=\|a\|^{2^k}$. Taking the spectral-radius limit along these powers gives
$$
\boxed{\nu_A(a)=\|a\|\quad\text{for normal }a.}
$$
Let $\varphi:A\to B$ be a <C-star homomorphism>, without assuming continuity or preservation of the ambient unit. Put $p=\varphi(1_A)$, a <projection in a C-star algebra>. If $p=0$, then $\varphi=0$. Otherwise its image lies in the unital corner $pBp$ and it is unital as a map into that corner. An inverse in $A$ maps to an inverse in the corner, so $\sigma_{pBp}(\varphi(b))\subseteq\sigma_A(b)$. Apply this to the normal element $b=a^*a$ and use the <C-star identity> in both algebras:
$$
\|\varphi(a)\|^2=\|\varphi(a^*a)\|=\nu_{pBp}(\varphi(a^*a))\leq\nu_A(a^*a)=\|a^*a\|=\|a\|^2.
$$
Thus \b[every <C-star homomorphism> is <norm> non-increasing], including nonunital maps.
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