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

TL;DR

This paper develops an index theory for projective families of elliptic pseudodifferential operators with decomposable Dixmier-Douady class, establishing the equality of topological and analytic indices in twisted K-theory.

Contribution

It introduces a new index theory for decomposable twisting classes, linking Azumaya bundles to smoothing operators and computing the twisted Chern character.

Findings

Topological and analytic indices coincide for decomposable cases.

The twisted Chern character is explicitly computed using a Chern-Weil type approach.

Azumaya bundles can be realized via smoothing operators in this setting.

Abstract

An index theory for projective families of elliptic pseudodifferential operators is developed when the twisting, i.e. Dixmier-Douady, class is decomposable. One of the features of this special case is that the corresponding Azumaya bundle can be realized in terms of smoothing operators. The topological and the analytic index of a projective family of elliptic operators both take values in the twisted K-theory of the parameterizing space. 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.