A Note on Kasparov Product and Duality

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Using Paschke-Higson duality, we can get a natural index pairing Kᵢ(A) × Kᵢ₊₁(DΦ) → Z (i=0,1) (mod2), where A is a separable C-algebra, and Φ 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 C₁ and S are KK-equivalent using the odd index pairing.