= Solution
The <Beurling–Gelfand spectral radius formula> states that for every element $a$ of a complex <Banach algebra>,
$$
\boxed{r(a)=\max_{\lambda\in\sigma(a)}|\lambda|
=\lim_{n\to\infty}\|a^n\|^{1/n}=\inf_{n\geq1}\|a^n\|^{1/n}.}
$$
For a <Hermitian element of a C-star algebra> $a=a^*$, the <C-star identity> gives $\|a^2\|=\|a\|^2$. Iterating,
$$
\|a^{2^m}\|=\|a\|^{2^m}.
$$
The subsequence $n=2^m$ in the spectral-radius limit therefore proves $\boxed{r(a)=\|a\|}$.
Let $\theta:A\to B$ be a unital <C-star homomorphism>. If $a-\lambda1$ is invertible in $A$, its inverse is sent to an inverse of $\theta(a)-\lambda1$ in $B$. Thus $\sigma_B(\theta(a))\subseteq\sigma_A(a)$ without assuming continuity of $\theta$. For Hermitian $a$, $\theta(a)$ is also Hermitian, so
$$
\|\theta(a)\|=r_B(\theta(a))\leq r_A(a)=\|a\|.
$$
For arbitrary $x\in A$, apply this inequality to $x^*x$ and the <C-star identity> in both algebras:
$$
\|\theta(x)\|^2=\|\theta(x)^*\theta(x)\|=\|\theta(x^*x)\|\leq\|x^*x\|=\|x\|^2.
$$
Hence $\boxed{\|\theta(x)\|\leq\|x\|}$ for every $x$.
Now assume $\theta$ is injective. For a fixed $x$, it is enough to prove norm preservation on $C=C^*(1,x^*x)$, a commutative unital <C-star algebra>: the preceding square-norm identity will then recover the norm of $x$. By the <Commutative Gelfand--Naimark theorem>, identify $C$ with $C(K)$. Let $D=\overline{\theta(C)}\subseteq B$. It is also a commutative unital <C-star algebra>, and its <character space> $\Phi_D$ is compact. Each $\chi\in\Phi_D$ restricts along $\theta$ to a character of $C(K)$, hence to evaluation at a point $q(\chi)\in K$. The map $q:\Phi_D\to K$ is continuous, so $L=q(\Phi_D)$ is closed.
If $L\ne K$, the <Urysohn lemma> provides a nonzero $f\in C(K)$ vanishing on $L$. Then $\chi(\theta(f))=f(q(\chi))=0$ for every $\chi\in\Phi_D$. Faithfulness of the <Gelfand transform> in $D$ gives $\theta(f)=0$, contradicting injectivity. Thus $q$ is onto. The isometric <Gelfand transform> of $D$ now gives
$$
\|\theta(f)\|=\sup_{\chi\in\Phi_D}|\chi(\theta(f))|
=\sup_{t\in K}|f(t)|=\|f\|\quad(f\in C).
$$
Finally,
$$
\|\theta(x)\|^2=\|\theta(x^*x)\|=\|x^*x\|=\|x\|^2,
\qquad\boxed{\|\theta(x)\|=\|x\|.}
$$
This proves that an <injective C-star homomorphism is isometric> without assuming its image is closed before norm preservation has been established.
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