With an action of on a -algebra and a skew-symmetric matrix , one can consider the Rieffel deformation of , which is a -algebra generated by the -smooth elements of with a new multiplication. The purpose of this article is to obtain explicit formulas for -theoretical quantities defined by elements of . We give an explicit realization of the Thom class in in any dimension and use it in the index pairings. For local index formulas we assume that there is a densely defined trace on , invariant under the action. When is odd, for example, we give a formula for the index of operators of the form , where is the operator of left Rieffel multiplication by an invertible element over the unitization of and is the projection onto the nonnegative eigenspace of a Dirac operator constructed from the action . The results are new also for the undeformed case . The construction relies on two approaches to Rieffel deformations in addition to Rieffel’s original one: Kasprzak deformation and warped convolution. We end by outlining potential applications in mathematical physics.
"Index pairings for -actions and Rieffel deformations." Kyoto J. Math. 59 (1) 77 - 123, April 2019. https://doi.org/10.1215/21562261-2018-0003