Spatial Algebras: Algebraic Foundations, Spectral Reconstruction, and Applications - Tapa blanda

Sinopsis

Spatial Algebras develops a new algebraic framework obtained by extending the complex plane with real idempotent directions while preserving the identity of each channel. The resulting structures are commutative and have an identity element, but they are generally nonassociative. Rather than treating nonassociativity merely as a defect, the book shows how the associator can be used to reconstruct spatial information lost under complex projection.

Beginning with the necessary foundations, the text first develops finite-dimensional spatial algebras and then extends the theory to infinite-dimensional families built over classical sequence spaces. Topics include multiplication operators, spectral theory, resolvents, functional calculus, coupled spectral representations, associator reconstruction, and Hilbertian geometry.

The theory is subsequently transported to bundles and dynamical systems, with discussions of projectors, connections, transfer between channels, curvature, symmetries, and bifurcation. The final part presents two advanced applications: Multichannel Spectral Geometry and a positive realization of Kaluza-Klein channels.

Designed for readers with an undergraduate degree in mathematics, the book is self-contained and requires no previous study of nonassociative algebras. Definitions are supported by conceptual preparation, detailed proofs, examples, counterexamples, graded exercises, and answers or hints. Clear warnings distinguish established results from open directions and delimit the mathematical scope of the physical applications.

This volume will be of interest to students, researchers, and instructors working in algebra, functional analysis, operator theory, spectral theory, differential geometry, and mathematical physics.

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