Journal of Noncommutative Geometry

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Volume 14, Issue 1, 2020, pp. 1–23
DOI: 10.4171/JNCG/357

Published online: 2020-02-11

The spectral localizer for even index pairings

Terry A. Loring[1] and Hermann Schulz-Baldes[2]

(1) The University of New Mexico, Albuquerque, USA
(2) Universität Erlangen-Nürnberg, Germany

Even index pairings are integer-valued homotopy invariants combining an even Fredholm module with a $K_0$-class specified by a projection. Numerous classical examples are known from differential and non-commutative geometry and physics. Here it is shown how to construct a finite-dimensional self-adjoint and invertible matrix, called the spectral localizer, such that half its signature is equal to the even index pairing. This makes the invariant numerically accessible. The index-theoretic proof heavily uses fuzzy spheres.

Keywords: Index pairings, fuzzy spheres, numerical K-theory

Loring Terry, Schulz-Baldes Hermann: The spectral localizer for even index pairings. J. Noncommut. Geom. 14 (2020), 1-23. doi: 10.4171/JNCG/357