Let $f$ be a continuous map of the closure $\overline{\Delta}$ of the open unit disc $\Delta$ of $\mathbb{C}$ into a unital associative Banach algebra $\mathcal{A}$, whose restriction to $\Delta$ is holomorphic, and which satisfies the condition whereby $0 \notin \sigma(f(z)) \subset \overline{\Delta}$ for all $z \in \Delta$ and $\sigma(f(z)) \subset \partial \Delta$ whenever $z \in \partial \Delta$ (where $\sigma(x)$ is the spectrum of any $x \in \mathcal{A}$). One of the basic results of the present paper is that $f$ is , that is to say, $\sigma(f(z))$ is then a compact subset of $\partial \Delta$ that does not depend on $z$ for all $z \in \overline{\Delta}$. This fact will be applied to holomorphic self-maps of the open unit ball of some $J^{*}$-algebra and in particular of any unital $C^{*}$-algebra, investigating some cases in which not only the spectra but the maps themselves are necessarily constant.