Families of Dirac operators, boundaries and the B-calculus

A version of the Atiyah-Patodi-Singer index theorem is proved for general families of Dirac operators on compact manifolds with boundary. The vanishing of the analytic index of the boundary family, in K-1 of the base, allows us to define, through an explicit trivialization, a smooth family of boundary conditions of generalized Atiyah-Patodi-Singer type. The calculus of b-pseudodifferential operators is then employed to establish the family index formula. A relative index formula, describing the effect of changing the choice of the trivialization, is also given. In case the boundary family is invertible the form of the index theorem obtained by Bismut and Cheeger is recovered.