Operator-valued pseudo-differential operators and the twisted index pairing

Abstract: The notion of pseudo-differential operators with coefficients in a continuous trace algebra over a manifold are introduced and their index theory is studied. The algebra of principal symbols in this calculus provides an abstract Poincar\'e dual to the continuous trace algebra. Index formulas for pseudo-differential operators twisted by a bundle on the opposite continuous trace algebra are obtained in terms of an Atiyah-Singer type index formula describing the twisted index pairing as index theory for elliptic operators.