On the Lifting of the Dirac Elements in the Higson-Kasparov Theorem
Abstract
In this thesis, we investigate the proof of the Baum-Connes Conjecture with Coefficients for a--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. -theory and -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 -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 -algebra of (real) Hilbert space becomes a --algebra naturally even when a group 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.
Keywords
Cite
@article{arxiv.1608.06375,
title = {On the Lifting of the Dirac Elements in the Higson-Kasparov Theorem},
author = {Shintaro Nishikawa},
journal= {arXiv preprint arXiv:1608.06375},
year = {2016}
}