Smooth crossed products of Rieffel's deformations

TL;DR

This paper proves that Rieffel's deformation preserves periodic cyclic cohomology by constructing an explicit isomorphism between the crossed products of the original and deformed algebras, simplifying the understanding of deformation invariance.

Contribution

It constructs an explicit isomorphism between the crossed products of a Frechet algebra and its Rieffel deformation, linking different deformation approaches and invariants.

Findings

Periodic cyclic cohomology remains invariant under Rieffel deformation.

Explicit isomorphism between crossed products of original and deformed algebras.

Simplified proof of equivalence between Rieffel's and Kasprzak's deformation methods.

Abstract

Assume A is a Frechet algebra equipped with a smooth isometric action of a vector group V, and consider Rieffel's deformation A_J of A. We construct an explicit isomorphism between the smooth crossed products V\ltimes\A_J and V\ltimes\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.