Letters in Mathematical Physics

, Volume 104, Issue 3, pp 361–371 | Cite as

Smooth Crossed Products of Rieffel’s Deformations



Assume \({\mathcal{A}}\) is a Fréchet algebra equipped with a smooth isometric action of a vector group V, and consider Rieffel’s deformation \({\mathcal{A}_J}\) of \({\mathcal{A}}\). We construct an explicit isomorphism between the smooth crossed products \({V\ltimes\mathcal{A}_J}\) and \({V\ltimes\mathcal{A}}\). When combined with the Elliott–Natsume–Nest isomorphism, this immediately implies that the periodic cyclic cohomology is invariant under deformation. Specializing to the case of smooth subalgebras of C*-algebras, we also get a simple proof of equivalence of Rieffel’s and Kasprzak’s approaches to deformation.

Mathematics Subject Classification

53D55 19D55 46L65 


Deformation quantization Rieffel’s deformation cyclic cohomology crossed products 


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsloOsloNorway

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