Spectrum in alternative topological algebras and a new look at old theorems

TL;DR

This work develops a unified spectral framework for left and right alternative topological algebras over real, complex, and general fields, and uses spectra to extend foundational division-algebra theorems to nonassociative settings. It proves that in quadratic algebras the notions of flexible and proper coincide, and introduces canonical bilinear forms and involutions that interact with algebra operations. By defining spectra for one-sided algebras, the authors derive hyperinvariant subspaces for operators on Fréchet spaces and establish topological versions of Frobenius, Zorn, Gelfand–Mazur, and Hurwitz-type results. In real and complex contexts they show the existence and uniqueness of norms compatible with power maps and demonstrate that several algebras are locally complex or division under natural conditions, thereby broadening classical theorems to broader nonassociative-topological settings. The paper thereby contributes tools for nonassociative spectral theory and highlights the role of local complex structure in unifying these results.

Abstract

In this paper, we consider real and complex algebras as well as algebras over general fields. In Section 2, we revisit and prove several results on (quadratic) algebras over general fields. As an example, we demonstrate that a quadratic algebra over a field of characteristic not $2$ is flexible if and only if it is proper--a concept introduced in this paper. In Section 3, we show how to develop the spectral theory in the context of complex (resp. real) one-sided alternative topological algebras. As an application of the existence of spectrum, we prove the existence of nontrivial hyperinvariant linear manifolds for nonscalar (resp. nonquadratic) continuous linear operators acting on complex (resp. real) Fréchet spaces. Along the way, spectral theory is used to prove several topological counterparts of the well-known theorems of Frobenius, Hurwitz, Gelfand-Mazur, and Zorn. This is done, for example, in the context of left (resp. right) alternative topological algebras whose duals separate their elements. In Section 4, we consider real and complex algebras in various topological settings and reconsider and prove several results. For instance, it is shown that given a $ 1 < k \in \mathbb{N}$, on any locally complex algebra, there exists a unique nonzero vector space norm, say, $\|.\|$, satisfying the identity $\|a^k\| = \|a\|^k$ on the algebra. In Section 5, among other things, we revisit and slightly strengthen the celebrated theorems of Frobenius, Zorn, Gelfand-Mazur, and Hurwitz, and also give slight extensions of their topological counterparts, e.g., theorems of Albert, Kaplansky, and Urbanik-Wright to name a few, in several settings.

Theorems & Definitions (31)

• Proposition 1.1
• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Proposition 2.9