The index of projective families of elliptic operators: The decomposable case

Abstract

An index theory for projective families of elliptic pseudodiffer-ential operators is developed under two conditions. First, that the twisting, i.e. Dixmier-Douady, class is in H² (Xℤ) ∪ H¹ (X; Z) ⊂ H³ (Zℤ) and secondly that the 2-class part is trivialized on the total space of the fibration. One of the features of this special case is that the corresponding Azumaya bundle can be refined to a bundle of smoothing operators. The topological and the analytic index of a projective family of elliptic operators associated with the smooth Azumaya bundle both take values in twisted K-theory of the parameterizing space and the main result is the equality of these two notions of index. The twisted Chern character of the index class is then computed by a variant of Chern-Weil theory.