Atiyah-Singer Dirac Operator on spacetimes with non-compact Cauchy hypersurface

Let $M$ be a globally hyperbolic manifold with complete spacelike Cauchy hypersurface $\Sigma \subset M$. Building on past and recent works of B\"ar and Strohmaier, we extend their Fredholm result of the Atiyah-Singer Dirac operator on compact Lorentzian spaces to the case, where $M$ is diffeomorphic to a product of $\Sigma$ with a compact time intervall and the hypersurface is a Galois covering with respect to a group $\Gamma$. We follow the first approach of both authors in this extended setting, where a well-posedness result of the Cauchy problem for the Dirac operator on non-compact manifolds is needed in preparation. After employing von Neumann algebras and further ingredients for Galois coverings, the well-posedness result is specified for the setting of interest, which leads to $\Gamma$-Fredholmness of the Dirac operator under APS boundary conditions.