Continuous spectrum-shrinking maps between finite-dimensional algebras

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be unital finite-dimensional complex algebras, each equipped with the unique Hausdorff vector topology. Denote by $\mathrm{Max}(\mathcal{A})=\{\mathcal{M}_1, \ldots, \mathcal{M}_p\}$ and $\mathrm{Max}(\mathcal{B})=\{\mathcal{N}_1, \ldots, \mathcal{N}_q\}$ the sets of all maximal ideals of $\mathcal{A}$ and $\mathcal{B}$, respectively. For each $1 \leq i \leq p$ and $1 \leq j \leq q$ define the quantities $$ k_i:=\sqrt{\dim(\mathcal{A}/\mathcal{M}_i)} \quad \text{ and } \quad m_j:=\sqrt{\dim(\mathcal{B}/\mathcal{N}_j)}, $$ 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. $\mathrm{sp}(\phi(a))\subseteq \mathrm{sp}(a)$ for all $a \in \mathcal{A}$) if and only if for each $1\leq j \leq q$ the linear Diophantine equation $$ k_1x_{1}^{j} + \cdots + k_px_{p}^j = m_j $$ has a non-negative integer solution $(x_{1}^j,\ldots,x_{p}^j)\in \mathbb{N}_{0}^p$. In a similar manner we also characterize the existence of continuous spectrum-preserving maps $\phi: \mathcal{A} \to \mathcal{B}$ (i.e. $\mathrm{sp}(\phi(a))= \mathrm{sp}(a)$ 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.