We give a uniform construction of the higher indices of elliptic operators associated to Alexander-Spanier cocycles of either parity in terms of a pairing between the $K$-theory and the cyclic cohomology of the algebra of complete symbols of pseudodifferential operators. This construction allows us to obtain an extension of Helton-Howe trace formulas. Further analysis of the cyclic theory of the algebra of complete symbols leads to construction of secondary invariants of pseudodifferential operators.
This is joint work with H. Moscovici.