  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.

K-Theory and Homology;Differential Geometry;Operator Algebras