Journal of Noncommutative Geometry


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Volume 11, Issue 3, 2017, pp. 887–917
DOI: 10.4171/JNCG/11-3-4

Published online: 2017-09-26

Boundary value problems with Atiyah–Patodi–Singer type conditions and spectral triples

Ubertino Battisti[1] and Joerg Seiler[2]

(1) Università di Torino, Italy
(2) Università di Torino, Italy

We study realizations of pseudodifferential operators acting on sections of vector-bundles on a smooth, compact manifold with boundary, subject to conditions of Atiyah–Patodi–Singer type. Ellipticity and Fredholm property, compositions, adjoints and self-adjointness of such realizations are discussed. We construct regular spectral triples $(\mathcal {A,H,D})$ for manifolds with boundary of arbitrary dimension, where $\mathcal H$ is the space of square integrable sections. Starting out from Dirac operators with APS-conditions, these triples are even in case of even dimensional manifolds; we show that the closure of $\mathcal A$ in $\mathcal L(\mathcal H)$ coincides with the continuous functions on the manifold being constant on each connected component of the boundary.

Keywords: Spectral triples, manifolds with boundary, boundary value problems with APS-type conditions, pseudodifferential operators

Battisti Ubertino, Seiler Joerg: Boundary value problems with Atiyah–Patodi–Singer type conditions and spectral triples. J. Noncommut. Geom. 11 (2017), 887-917. doi: 10.4171/JNCG/11-3-4