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We show that the Thorn isomorphism and the Pimsner-Voiculescu exact sequence both hold for smooth crossed products of Fréchet algebras by      R \mathbb {R}       and      Z \mathbb {Z}       respectively. We also obtain the same results for      L 1 {L^1}      -crossed products of Banach algebras by      R \mathbb {R}       and      Z \mathbb {Z}      .

Representable 𝐾-theory of smooth crossed products by 𝑅 and 𝑍