Index pairings for R n -actions and Rieffel deformations
With an action α of Rn on a C∗-algebra A and a skew-symmetric n×n matrix Θ, one can consider the Rieffel deformation AΘ of A, which is a C∗-algebra generated by the α-smooth elements of A with a new multiplication. The purpose of this article is to obtain explicit formulas for K-theoretical quantities defined by elements of AΘ. We give an explicit realization of the Thom class in KK in any dimension n and use it in the index pairings. For local index formulas we assume that there is a densely defined trace on A, invariant under the action. When n is odd, for example, we give a formula for the index of operators of the form PπΘ(u)P, where πΘ(u) is the operator of left Rieffel multiplication by an invertible element u over the unitization of A and P is the projection onto the nonnegative eigenspace of a Dirac operator constructed from the action α. The results are new also for the undeformed case Θ=0. 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.