2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/226681We define generalized Atiyah-Patodi-Singer boundary conditions of product type for Dirac operators associated to C*-vector bundles on the product of a compact manifold with boundary and a closed manifold. We prove a product formula for the K-theoretic index classes, which we use to generalize the product formula for the topological signature to higher signatures.33 pages; revised and expanded, for the odd signature class more general definition (depending on a Lagrangian) usedDifferential GeometryK-Theory and Homology58J32;19K56;57R20text