## Kyoto Journal of Mathematics

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

Andreas Andersson

#### 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 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 $P\pi^{\Theta}(u)P$, where $\pi^{\Theta}(u)$ is the operator of left Rieffel multiplication by an invertible element $u$ over the unitization of $A$ and $P$ is 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.

#### Article information

Source
Kyoto J. Math., Volume 59, Number 1 (2019), 77-123.

Dates
Received: 15 November 2016
Revised: 28 December 2016
Accepted: 28 December 2016
First available in Project Euclid: 23 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1534989636

Digital Object Identifier
doi:10.1215/21562261-2018-0003

Mathematical Reviews number (MathSciNet)
MR3934624

Zentralblatt MATH identifier
07081623

#### Citation

Andersson, Andreas. Index pairings for $\mathbb{R}^{n}$ -actions and Rieffel deformations. Kyoto J. Math. 59 (2019), no. 1, 77--123. doi:10.1215/21562261-2018-0003. https://projecteuclid.org/euclid.kjm/1534989636

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