The spectral localizer for even index pairings. (English) Zbl 1441.19011
Summary: 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.
MSC:
| 19K56 | Index theory |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 58J28 | Eta-invariants, Chern-Simons invariants |
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