Collapse in noncommutative geometry and spectral continuity

Abstract In the classical realm of Riemannian geometry, informally, if two manifolds are close in the Gromov–Hausdorff distance, and belong to a class of closed manifolds with bounded curvature and diameter, then the spectra of their Laplacian or Dirac operators are also close under many scenarii. Of particular interest is the case where a sequence of manifolds converges for the Gromov–Hausdorff distance to a manifold of lower dimension, that is, collapses to the limit. The question then arises of the continuity, in some sense, of the geometrically relevant operators and their spectra. This has been investigated at length in many classical papers. In the more general context of noncommutative geometry, the notion of convergence with respect to the variation of certain metric structures plays a very important role, as it allows the approximation of spaces and algebras over Hilbert spaces by simpler ones (e.g., finite‐dimensional and/or matricial). So, if two quantum compact metric spaces are close in the metric sense, then how similar are they, as noncommutative spaces? In this paper, we initiate the study of the continuity of spectra and other properties of metric spectral triples on noncommutative ‐bundles, for compact Lie, under collapse in the “vertical” direction. As a first step in this study, we work with the spectral propinquity , an analog of the Gromov–Hausdorff distance for metric spectral triples introduced by the second author. The spectral propinquity is a form of metric for differential structures. Inspired by results from collapse in Riemannian geometry, we study spectral triples that decompose, in some sense, in a vertical and a horizontal direction; they can be shown to be special Kasparov products. We perturb the (metric in the) vertical component by a parameter , and then we take the limit for approaching zero, thus obtaining a metric spectral triples convergence results. As a consequence, by the work of the second author, we also derive continuity results for the spectra of the Dirac operators of these spectral triples. Examples of applications of our work include collapse of products of spectral triples with one Abelian factor, spectral triples associated to principal ‐bundles over closed Riemannian spin manifolds, and spectral triples associated to noncommutative principal ‐bundles as in the work of Schwieger and Wagner; include ‐crossed products.