Geometric Structure Spaces

RESEARCH PURPOSE AND SCOPE

This project studies -algebras from the viewpoint of Banach-space geometry, with particular emphasis on characterizing the algebras of all compact operators on complex Hilbert spaces. The central tool is the geometric structure space, a closure space built from the geometry of the norm unit ball. It serves as a common superstructure for the irreducible spectrum and the primitive ideal space: both can be recovered through canonical constructions. The theory connects this geometric information to irreducible representations, compact-operator models, decomposition and nonlinear rigidity. The arXiv manuscript presents the mathematical development, while the LFH Companion provides a dependency-guided account of the formalized results and their exact Lean sources.

The geometric structure space is used as a closure space; it is not assumed to be a topological space. The -algebra results retain their individual representation, nonzero and continuity hypotheses. In particular, characterizing the whole algebra is distinct from asking whether an algebra has a faithful compact representation. The LFH Companion and its Cards state the precise assumptions for each characterization and rigidity result.

Recoverability through canonical closure-space constructions does not identify the two spectra as literal subspaces or assert a complete classification of arbitrary algebras.