Approximate Noncommutative Curvature, Index Theory, and Pseudodifferential Calculus on Non-Separable Banach Spaces

Abstract

Juan Alberto Molina Garcia

This article develops an analytic and geometric framework for approximate noncommutative curvature and index theory on non-separable Banach spaces (NSBS), extending classical operator-theoretic and pseudodifferential methods beyond the separability barrier. Motivated by the structural limitations of Banach spaces without countable Schauder bases, we construct a theory of approximate pseudodifferential operators (APDOs), defined through localised symbolic calculi on admissible nets rather than countable charts. A generalised approximate connection is introduced for operator modules over Banach–function algebras, enabling the definition of curvature-type quantities via commutator densities, asymptotic limits, and weakened continuity hypotheses compatible with non-separability.

The main contribution is a comprehensive formulation of an approximate index theorem for Fredholm-type operators in NSBS, based on spectral regularisation, approximate traces, and an extension of Connes’ noncommutative index pairing to settings devoid of σ-compactness and separability. We prove an Approximate Curvature–Index Correspondence, establishing that the analytic index of an APDO can be reconstructed from the asymptotic curvature density of chosen approximate connections. Applications include: (i) classification of curvature invariants for operator families on l ∞, C(K)with Knon-metrisable, and direct sums of non-separable Banach spaces; (ii) construction of parametric families for APDOs when classical pseudodifferential techniques fail; and (iii) an extension of the Wodzicki-type residue to a finitely additive non-separable trace.

This work advances the theory of noncommutative geometry towards infinite-dimensional, non-metrisable, and non- separable analytic structures, opening pathways towards curvature, index, and quantisation theories in the absence of standard countability assumptions.