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

We study realizations of pseudodifferential operators acting on sections of vectorbundles 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 (A;H;D) for manifolds with boundary of arbitrary dimension, where 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 A in L(H) coincides with the continuous functions on the manifold being constant on each connected component of the boundary.