Noncommutative elliptic theory. Examples

TL;DR

This paper develops index formulas for differential operators with noncommutative algebra coefficients, focusing on crossed products and noncommutative tori, advancing understanding in noncommutative elliptic theory.

Contribution

It introduces new index formulas for operators twisted by projections over crossed product algebras, including noncommutative tori, expanding noncommutative elliptic theory.

Findings

Index formulas for Euler, signature, and Dirac operators derived.

Index of Connes operators on the noncommutative torus computed.

Provides examples illustrating noncommutative elliptic operators.

Abstract

We study differential operators, whose coefficients define noncommutative algebras. As algebra of coefficients, we consider crossed products, corresponding to action of a discrete group on a smooth manifold. We give index formulas for Euler, signature and Dirac operators twisted by projections over the crossed product. Index of Connes operators on the noncommutative torus is computed.