Spatial Algebras: Algebraic Foundations, Spectral Reconstruction, and Applications - Paperback

by Kleber Câmara (Author)

Spatial Algebras develops a self-contained theory of commutative, unital, and generally nonassociative algebras that extend the complex numbers by adjoining real idempotent directions.

The book begins with finite-dimensional spatial algebras and introduces their algebraic structure, associators, idempotents, multiplication operators, elementary functions, spectral coordinates, and functional calculus. It then develops an infinite-dimensional scale of spatial algebras, with detailed treatments of bounded multiplication operators, spectra, resolvents, reconstruction through associators, and the corresponding Hilbertian theory.

The later chapters transport the algebraic decomposition to bundles, connections, curvature, dynamical systems, variational principles, symmetries, and bifurcation theory. Two advanced applications are presented separately: Multichannel Spectral Geometry and a heat-kernel-regularized realization of Kaluza-Klein channels. The mathematical scope and physical limitations of these applications are explicitly stated.

Designed for advanced undergraduate students, graduate students, instructors, and researchers, the book includes definitions, complete proofs, examples, counterexamples, graded exercises, and an extensive chapter of answers and hints. It is suitable for readers interested in algebra, functional analysis, operator theory, spectral theory, differential geometry, and mathematical physics.