关于Higson-Kasparov定理中Dirac元的提升
On the Lifting of the Dirac Elements in the Higson-Kasparov Theorem
摘要：
本文研究了α- $T$ -可列群的系数Baum-Connes猜想的证明。我们将主要和基本上遵循N. Higson和G.卡斯帕罗夫在论文中[奈杰尔·希格森和根纳迪·卡斯帕罗夫。对在希尔伯特空间上恰当地等距作用的群的$E$ -理论和$KK$ -理论。发明。数学，144（1）：23-74，2001]。关键特征如下。他们证明中的一个重要问题是如何求出等变$KK$ -理论中的Dirac元（Bott元的逆元）。我们证明了用于提升Dirac元的群同态在我们感兴趣的情况下是同构。从而对Higson-Kasparov定理中Dirac元的提升有了一个清晰而简单的理解。另一方面，在研究过程中，我们指出了一个问题，并给出了本文所定义的非交换谓词演算的一个固定的精确定义。我们提到（实）希尔伯特空间的$C^*$ -代数变成了$G$ - $C^*$ -代数自然，即使当一个群$G$作用在希尔伯特空间上的仿射作用，其线性部分的形式是等距乘以标量，并证明无限维Bott-周期性在这种情况下，通过使用费尔的吸收技术。
英文摘要：
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.
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