On the Lifting of the Dirac Elements in the Higson-Kasparov Theorem

TL;DR

This paper clarifies the lifting process of Dirac elements in the Higson-Kasparov proof of the Baum-Connes Conjecture, providing a simplified understanding and addressing technical issues in the non-commutative functional calculus.

Contribution

It proves the isomorphism of the group homomorphism used for lifting Dirac elements, offering a clearer perspective on the Higson-Kasparov theorem and fixing a definition in non-commutative functional calculus.

Findings

The group homomorphism for lifting Dirac elements is an isomorphism.

A fixed, precise definition for non-commutative functional calculus is provided.

The $C^*$-algebra of a Hilbert space is naturally a $G$-$C^*$-algebra under affine actions.

Abstract

In this thesis, we investigate the proof of the Baum-Connes Conjecture with Coefficients for a-$T$-menable groups. We will mostly and essentially follow the argument employed by N. Higson and G. Kasparov in the paper [Nigel Higson and Gennadi Kasparov. $E$-theory and $KK$-theory for groups which act properly and isometrically on Hilbert space. Invent. Math., 144(1):23-74, 2001]. The crucial feature is as follows. One of the most important point of their proof is how to get the Dirac elements (the inverse of the Bott elements) in Equivariant $KK$-Theory. We prove that the group homomorphism used for the lifting of the Dirac elements is an isomorphism in the case of our interests. Hence, we get a clear and simple understanding of the lifting of the Dirac elements in the Higson-Kasparov Theorem. In the course of our investigation, on the other hand, we point out a problem and give a fixed precise definition for the non-commutative functional calculus which is defined in the paper In the final part, we mention that the $C^*$-algebra of (real) Hilbert space becomes a $G$-$C^*$-algebra naturally even when a group $G$ acts on the Hilbert space by an affine action whose linear part is of the form an isometry times a scalar and prove the infinite dimensional Bott-Periodicity in this case by using Fell's absorption technique.