Index pairings for $\mathbb{R}^n$-actions and Rieffel deformations

Abstract

With an action $\alpha$ of $\mathbb{R}^n$ on a $C^*$-algebra $A$ and a skew-symmetric $n\times n$ matrix $\Theta$ one can consider the Rieffel deformation $A_\Theta$ of $A$, which is a $C^*$-algebra generated by the $\alpha$-smooth elements of $A$ with a new multiplication. The purpose of this paper is to obtain explicit formulas for $K$-theoretical quantities defined by elements of $A_\Theta$. We assume that there is a densely defined trace on $A$, invariant under the action. We give an explicit realization of Thom class in $KK$ in any dimension $n$, and use it in the index pairings. When $n$ is odd, for example, we give a formula for the index of operators of the form $P\pi^\Theta(u)P$, where $\pi^\Theta(u)$ is the operator of left Rieffel multiplication by an invertible element $u$ over the unitization of $A$, and $P$ is projection onto the nonnegative eigenspace of a Dirac operator constructed from the action $\alpha$. The results are new also for the undeformed case $\Theta=0$. 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.