Approximate Spectral Triples and Non-commutative Geometric Structures on Non-Separable Banach Spaces
Abstract
Juan Alberto Molina Garcia
This article develops a geometric extension of the approximate C^*- and W^*-algebra framework introduced in earlier work, showing that NSBS naturally give rise to noncommutative geometric structures in the sense of Connes. While classical noncommutative geometry is traditionally based on separable C^*-algebras and Hilbert spaces, many analytically relevant settings—such as l^∞, L^∞ (μ) for non-σ-finite measures, or C(βN)—lack separability and fall outside the scope of the usual spectral triple framework. The present work overcomes these limitations by developing a theory of approximate spectral triples, constructed as inductive limits of local spectral triples on separable components.
Given a non-separable Banach space X, its approximate operator algebra A^approx (X) is defined as an inductive limit of separable C^*-algebras A_F. We show that this structure canonically admits a family of densely defined local Dirac-type operators D_F acting on separable Hilbert modules H_F, satisfying compatibility conditions that allow the construction of a global approximate Dirac operator D^approx. The pair (A^approx,D^approx) behaves like a geometric object: it induces a variational metric on the state space, determines a quantum differential calculus, and provides a platform for an approximate version of Connes’ distance formula.
We prove that approximate spectral triples endow A^approx with the structure of a weakly differentiable noncommutative manifold, with the following key features: 1) an approximate spectral metric d^approx defined on approximate states; 2) an approximate Calderón–Zygmund decomposition on separable subspaces; 3) a differential calculus generated by commutators [D_F,a_F], compatible through the inductive system; 4) an associated approximate K-homology and K-theory enabling topological classification; 5) explicit geometric models on benchmark EBNS: l^∞, C(βN), and L^∞ (μ) with non-σ-finite μ.
This shows that non-separable Banach spaces carry a noncommutative geometric structure, expressible through approximate spectral data, and exhibiting differential regularity only locally. This yields a rigorous notion of non-separable quantum manifold: a geometric space whose differential, metric, and spectral structures are visible only through separable windows.
To our knowledge, this approach is entirely new and opens a path for applying noncommutative geometry to non-separable functional analysis, mathematical physics, and higher operator theory.