= Solution
For a unital <Banach algebra>, $\varphi(a)$ belongs to $\sigma_A(a)$: otherwise $a-\varphi(a)1$ would be invertible, while applying $\varphi$ to its inverse identity would give $0=1$. Therefore
$$
|\varphi(a)|\leq r(a)\leq\lVert a\rVert,
$$
so every <character of an algebra> is continuous and has norm one. The nonunital case follows by extending the character to the <unitization of an algebra>.
The <Gelfand topology> on $\Phi_A$ is the <weak-star topology> inherited from $A^*$: a net $\varphi_i$ converges to $\varphi$ exactly when $\varphi_i(a)\to\varphi(a)$ for every $a\in A$. If $A$ is unital, $\Phi_A$ lies in the weak-star compact dual unit ball by the <Banach-Alaoglu theorem>. The equations
$$
\varphi(1)=1,
\qquad
\varphi(ab)=\varphi(a)\varphi(b)
$$
define a weak-star closed subset, so $\Phi_A$ is compact.
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