A Note on Kasparov Product and Duality

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.

💡 Research Summary

The paper investigates the relationship between Paschke‑Higson duality and the Kasparov product, establishing that a natural index pairing arising from the duality is in fact a special case of the Kasparov product. Let (A) be a separable (C^{*})-algebra and (\Phi\colon A\to\mathcal B(H)) a representation on a separable infinite‑dimensional Hilbert space (H). The Paschke‑Higson construction yields the “dual” algebra
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