  Using Paschke-Higson duality, we can get a natural index pairing $K_{i}(A)
\times K_{i+1}(D_{\Phi}) \to \boldsymbol{Z} \quad (i=0,1) (\mbox{mod}2)$, where
$A$ is a separable $C\sp*$-algebra, and $\Phi$ is a representation of $A$ on a
separable infinite dimensional Hilbert space $H$. It is proved that this is a
special case of the Kasparov Product. As a step, we show a proof of
Bott-periodicity for KK-theory asserting that $\mathbb{C}_1$ and $S$ are
$KK$-equivalent using the odd index pairing.

Operator Algebras;K-Theory and Homology