Continuous spectrum-shrinking maps between finite-dimensional power-associative algebras

Abstract

For finite-dimensional complex power-associative algebras $\mathcal{A}$ and $\mathcal{B}$, each equipped with its unique Hausdorff vector topology, we characterize continuous spectrum-shrinking maps $ϕ:\mathcal{A}\to\mathcal{B}$, i.e. maps satisfying $\mathrm{sp}_{\mathcal{B}}(ϕ(x)) \subseteq\mathrm{sp}_{\mathcal{A}}(x)$ for all $x\in\mathcal{A}$. The spectrum of an element is the zero set of its minimal polynomial, computed in the unitization for nonunital algebras. No such map exists when $\mathcal{A}$ is unital and $\mathcal{B}$ is nonunital. Otherwise, we symmetrize both products by setting $x\circ y:=\tfrac{1}{2}(xy+yx)$ and factor out the nilradicals to obtain semisimple Jordan quotients. We prove that a continuous map $ϕ$ is spectrum-shrinking precisely when, for every $x\in\mathcal{A}$, Jacobson's generic minimal polynomials of the simple quotient coordinates of $ϕ(x)$ are products of fixed nonnegative integer powers of those of $x$, with an additional fixed monomial factor allowed when $\mathcal{A}$ is nonunital. Comparing degrees gives linear Diophantine equations characterizing existence, and every nonnegative integer solution is realized by a polynomial spectrum-shrinking map. Spectrum preservation, meaning equality of spectra, is equivalent to additional positivity conditions. We also determine when every continuous spectrum-shrinking map preserves spectra and show that every continuous map $f:\mathcal{A}\to\mathbb{C}$ with $f(x)\in\mathrm{sp}_{\mathcal{A}}(x)$ for all $x\in\mathcal{A}$ is $\mathbb{C}$-linear and satisfies $f(x\circ y)=f(x)f(y)$ for all $x,y\in\mathcal{A}$.