Kyoto Journal of Mathematics

Index pairings for Rn-actions and Rieffel deformations

Andreas Andersson

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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.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19K56: Index theory [See also 58J20, 58J22]
Secondary: 19K33: EXT and $K$-homology [See also 55N22] 19K35: Kasparov theory ($KK$-theory) [See also 58J22]

Thom isomorphism Kasparov KK-theory spectral triples noncommutative geometry


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.

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