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 for manifolds with boundary of arbitrary dimension, where 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 in coincides with the continuous functions on the manifold being constant on each connected component of the boundary.
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Ubertino Battisti, Joerg Seiler, Boundary value problems with Atiyah–Patodi–Singer type conditions and spectral triples. J. Noncommut. Geom. 11 (2017), no. 3, pp. 887–917DOI 10.4171/JNCG/11-3-4