Non-Separable Spectral K-Theory: Homotopy Invariants for Operators Beyond Separability

This project constructs a systematic algebraic-topological framework for operator theory on non-separable Banach spaces, addressing gaps where classical spectral theory and K-theory rely on separability. Inspired by Novikov’s K-theory program, it aims to build classifying spaces for bundles of non-separable Banach algebras/operators and define K-groups and spectral-flow-like invariants adapted to non-separable contexts. The goals include axiomatizing a stable homotopy category for non-separable Banach modules, characterizing persistence of Fredholm indices and Bott periodicity, and defining new obstructions detecting "wild" spectral components unique to non-separability. This bridges functional analysis with modern homotopy theory, impacting operator algebras, PDEs on large state spaces, infinite-dimensional control, and providing a clean language for generalized spectral theorems suggested by Molina García.

References:

1. NEW IDEAS IN ALGEBRAIC TOPOLOGY ($K$-THEORY AND ITS APPLICATIONS). S. Novikov (1965).
2. Spectral Theory in Non-Separable Banach Spaces: Generalizations, New Properties and Structural Extension. Juan Alberto Molina García (2025). Thermodynamics Research: Open Access.