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April 2019 Index pairings for Rn-actions and Rieffel deformations
Andreas Andersson
Kyoto J. Math. 59(1): 77-123 (April 2019). DOI: 10.1215/21562261-2018-0003


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.


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Andreas Andersson. "Index pairings for Rn-actions and Rieffel deformations." Kyoto J. Math. 59 (1) 77 - 123, April 2019.


Received: 15 November 2016; Revised: 28 December 2016; Accepted: 28 December 2016; Published: April 2019
First available in Project Euclid: 23 August 2018

zbMATH: 07081623
MathSciNet: MR3934624
Digital Object Identifier: 10.1215/21562261-2018-0003

Primary: 19K56
Secondary: 19K33 , 19K35

Keywords: Kasparov KK-theory , noncommutative geometry , spectral triples , Thom isomorphism

Rights: Copyright © 2019 Kyoto University


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Vol.59 • No. 1 • April 2019
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