A K-homological approach to the quantization commutes with reduction problem

TL;DR

This paper develops a K-homological framework for the quantization commutes with reduction theorem, utilizing Kasparov's Dirac element and localization formulas in the context of crossed product C*-algebras.

Contribution

It introduces a novel K-homological approach to the quantization problem, extending the theory through localization formulas and Dirac elements.

Findings

Proved a localization formula for the Dirac element in K-homology.

Defined quantization of Hamiltonian G-spaces via push-forward of the Dirac element.

Established a K-homological proof of the quantization commutes with reduction theorem.

Abstract

Kasparov defined a distinguished K-homology fundamental class, so called the Dirac element. We prove a localization formula for the Dirac element in K-homology of crossed product of C^{*}-algebras. Then we define the quantization of Hamiltonian G-spaces as a push-forward of the Dirac element. With this, we develop a K-homological approach to the quantization commutes with reduction theorem.