  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.
