We define smooth generalized crossed products and prove six-term exact sequences of Pimsner–Voiculescu type. This sequence may, in particular, be applied to smooth subalgebras of the quantum Heisenberg manifolds in order to compute the generators of their cyclic cohomology. Further, our results include the known results for smooth crossed products. Our proof is based on a combination of arguments from the setting of (Cuntz–)Pimsner algebras and the Toeplitz proof of Bott periodicity.
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Olivier Gabriel, Martin Grensing, Six-term exact sequences for smooth generalized crossed products. J. Noncommut. Geom. 7 (2013), no. 2, pp. 499–524DOI 10.4171/JNCG/124