Continuous spectrum-shrinking maps between finite-dimensional power-associative algebras
Abstract
For finite-dimensional complex power-associative algebras 𝒜 and ℬ, each equipped with its unique Hausdorff vector topology, we characterize continuous spectrum-shrinking maps ϕ:𝒜→ℬ, i.e. maps satisfying spℬ(ϕ(x)) ⊆sp𝒜(x) for all x∈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 𝒜 is unital and ℬ is nonunital. Otherwise, we symmetrize both products by setting x∘ y:=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∈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 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:𝒜→ℂ with f(x)∈sp𝒜(x) for all x∈𝒜 is ℂ-linear and satisfies f(x∘ y)=f(x)f(y) for all x,y∈𝒜.
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BibTeX
@misc{gogic2026continuous,
title = {{Continuous spectrum-shrinking maps between finite-dimensional power-associative algebras}},
author = {Ilja Gogić},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29393},
eprint = {2610.29393},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29393},
}
RIS
TY - UNPB
TI - Continuous spectrum-shrinking maps between finite-dimensional power-associative algebras
AU - Ilja Gogić
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29393
UR - https://arxiv.org/abs/2610.29393
AB - For finite-dimensional complex power-associative algebras 𝒜 and ℬ, each equipped with its unique Hausdorff vector topology, we characterize continuous spectrum-shrinking maps ϕ:𝒜→ℬ, i.e. maps satisfying sp_ℬ(ϕ(x)) ⊆sp_𝒜(x) for all x∈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 𝒜 is unital and ℬ is nonunital. Otherwise, we symmetrize both products by setting x∘ y:=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∈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 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:𝒜→ℂ with f(x)∈sp_𝒜(x) for all x∈𝒜 is ℂ-linear and satisfies f(x∘ y)=f(x)f(y) for all x,y∈𝒜.
ER -