2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/125600We introduce a new topology, weaker than the gap topology, on the space of selfadjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of selfadjoint Breuer-Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips' analytical definition and derive integral formulas for the spectral flow for certain paths of unbounded operators with common domain, generalizing those of Carey--Phillips. Furthermore we prove the homotopy invariance of the real-valued index. As an example we consider invariant symmetric elliptic differential operators on Galois coverings.26 pages; revised and extended version, section on integral formulas addedOperator AlgebrasFunctional Analysis58J30; 58J22; 46C15text