Smooth Crossed Products of Rieffel’s Deformations

Assume A is a Fréchet algebra equipped with a smooth isometric action of a vector group V , and consider Rieffel’s deformation AJ of A. We construct an explicit isomorphism between the smooth crossed products V n AJ and V n 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. Introduction The main goal of this note is to give a short proof of invariance of periodic cyclic cohomology under Rieffel’s deformations. Particular cases of this result are, of course, well-known. For the noncommutative 2-tori this was already shown by Connes in the foundational paper [1]. The result was extended to the higher dimensional noncommutative tori by Nest [9]. More recently, similar results have been obtained for θ-deformations by Yamashita [12] and Sangha [11]. A possible systematic way of approaching the question of invariance of periodic cyclic (co)homology is by using the Gauss-Manin connection, see e.g. [13, 14], but in the analytic setting this usually involves significant technical difficulties. It is often more efficient to use crossed product decompositions. Given a Fréchet algebra A with a smooth isometric action of a vector group V , for Rieffel’s deformation AJ of A we construct an isomorphism between the smooth crossed products V n AJ and V nA. The existence of such an isomorphism on the C∗-algebra level is known [7, 4], but the proof of this existence has been rather indirect and relied heavily on the C∗-algebra technique. As it turns out, the origin of this isomorphism could not be easier: both smooth crossed products are naturally represented on the space S(V ;A) of A-valued Schwartz functions on V , and their images under these representations coincide. The isomorphism V nAJ ∼= V nA gives an embedding of AJ into the multiplier algebra of V nA. For θ-deformations a formula for such an embedding in terms of the decomposition of A into spectral subspaces is easy to guess, which was already used in the work of Connes and Landi [2]. For general Rieffel’s deformations, when there are no nonzero spectral subspaces, it is impossible to write down such a formula, yet the isomorphism V nAJ ∼= V nA has an explicit and relatively simple form. In the second part of this note we consider smooth subalgebras A ⊂ A of C∗-algebras. For C∗algebras, a different approach to deformation has been proposed by Kasprzak [7]. In his approach the existence of an isomorphism V nAJ ∼= V nA is taken as part of the definition of AJ , so that AJ is from the beginning defined as a subalgebra of M(V n A). Concretely, elements of AJ can be obtained using either Landstad’s theory [7] or certain quantization maps A → M(V n A) [4, 8]. Equivalence of two approaches has been proved in [4], but the proof was far from straightforward. Using our explicit isomorphism V nAJ ∼= V nA we can now give a very simple proof. To complete the picture, we also describe the quantization maps A→M(V nA) in Rieffel’s setting. Date: July 8, 2013; minor changes November 14, 2013. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 307663. 1