Actions of compact groups, C*-index theorem, and families

TL;DR

This paper proves an index theorem for elliptic operators on bundles with fibers as projective modules over C*-algebras, considering compact Lie group actions, and extends it to equivariant families over parameter spaces.

Contribution

It introduces a new index theorem for elliptic operators with fibers as projective modules over C*-algebras under compact group actions, extending classical results.

Findings

Established the index theorem for elliptic operators with C*-algebraic fibers.

Derived the equivariant index theorem for families over product spaces.

Connected group actions with index theory in a new algebraic setting.

Abstract

We prove the index theorem for elliptic operators acting on sections of bundles where fiber is equal to a projective module over a C*-algebra, in the situation of action of a compact Lie group on this algebra as well as on the total space commuting with symbol. As an application the equivariant index theorem for families over the direct product of base by the space of parameters is obtained.