A Note on Kasparov Product and Duality

TL;DR

This paper explores the relationship between Paschke-Higson duality and Kasparov product, demonstrating a natural index pairing in KK-theory and proving Bott-periodicity via the odd index pairing.

Contribution

It establishes that the index pairing derived from Paschke-Higson duality is a special case of the Kasparov product and provides a new proof of Bott-periodicity in KK-theory.

Findings

Index pairing is a special case of Kasparov product.

Proved Bott-periodicity using odd index pairing.

Established KK-equivalence of _1 and S.

Abstract

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.