Continuous spectrum-shrinking maps between finite-dimensional algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital finite-dimensional complex algebras, each equipped with the unique Hausdorff vector topology. Denote by $\mathrm{M}\mathrm{a}\mathrm{x}(\mathcal{A})=\{{\mathcal{M}}_1,\dots ,{\mathcal{M}}_p\}$ and $\mathrm{M}\mathrm{a}\mathrm{x}(\mathcal{B})=\{{\mathcal{N}}_1,\dots ,{\mathcal{N}}_q\}$ the sets of all maximal ideals of $\mathcal{A}$ and $\mathcal{B}$, respectively. For each $1\le i\le p$ and $1\le j\le q$ define the quantities $$k_i:=\sqrt{\dim (\mathcal{A}\mathrm{/}{\mathcal{M}}_i)}\text{ and }{m}_j:=\sqrt{\dim (\mathcal{B}\mathrm{/}{\mathcal{N}}_j)},$$ which are positive integers by Wedderburn’s structure theorem. We show that there exists a continuous spectrum-shrinking map $\phi :\mathcal{A}\to \mathcal{B}$ (i.e. $\mathrm{s}\mathrm{p}(\phi (a))\subseteq \mathrm{s}\mathrm{p}(a)$ for all $a\in \mathcal{A}$) if and only if for each $1\le j\le q$ the linear Diophantine equation $$k_1{x}_1^j+\cdots +k_p{x}_p^j=m_j$$ has a non-negative integer solution $(x_1^j,\dots ,x_p^j)\in {\mathbb{N}}_0^p$. In a similar manner we also characterize the existence of continuous spectrum-preserving maps $\phi :\mathcal{A}\to \mathcal{B}$ (i.e. $\mathrm{s}\mathrm{p}(\phi (a))=\mathrm{s}\mathrm{p}(a)$ for all $a\in \mathcal{A}$). Finally, we analyze conditions under which all continuous spectrum-shrinking maps $\phi :\mathcal{A}\to \mathcal{B}$ are automatically spectrum-preserving.