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 $\mathrm{Max}(\mathcal{A})=\{{\mathcal{M}}_1,\dots ,{\mathcal{M}}_p\}$ and $\mathrm{Max}(\mathcal{B})=\{{\mathcal{N}}_1,\dots ,{\mathcal{N}}_q\}$ 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 $$k_i:=\sqrt{\dim (\mathcal{A}/{\mathcal{M}}_i)}\text{ and }{m}_j:=\sqrt{\dim (\mathcal{B}/{\mathcal{N}}_j)},$$ which are positive integers by Wedderburn's structure theorem. We show that there exists a continuous spectrum-shrinking map $\ddot{I}•:\mathcal{A}\to \mathcal{B}$ (i.e. $\mathrm{sp}(\ddot{I}•(a))\subseteq \mathrm{sp}(a)$ 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 $\ddot{I}•:\mathcal{A}\to \mathcal{B}$ (i.e. $\mathrm{sp}(\ddot{I}•(a))=\mathrm{sp}(a)$ for all $a\in \mathcal{A}$). Finally, we analyze conditions under which all continuous spectrum-shrinking maps $\ddot{I}•:\mathcal{A}\to \mathcal{B}$ are automatically spectrum-preserving.

8 pages, updated version of Theorem 1.1 (corrected from previous submission), to appear in J. Algebra Appl