Index pairings for Rn-actions and Rieffel deformations

Abstract

With an action $\alpha$ of ${\mathbb{R}}^n$ on a $C^{\ast }$-algebra $A$ and a skew-symmetric $n\times n$ matrix $\Theta$, one can consider the Rieffel deformation $A_{\Theta }$ of $A$, which is a $C^{\ast }$-algebra generated by the $\alpha$-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 $A_{\Theta }$. We give an explicit realization of the Thom class in $\mathit{KK}$ 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 $P{\pi }^{\Theta }\left(u\right)P$, where ${\pi }^{\Theta }\left(u\right)$ 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 $\alpha$. 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.