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Bivariant K-theory with R/Z-coefficients and rho classes of unitary representations
- We construct equivariant KK-theory with coefficients in and R/Z as suitable inductive limits over II1-factors. We show that the Kasparov product, together with its usual functorial properties, extends to KK-theory with real coefficients. Let Gamma be a group. We define a Gamma-algebra A to be K-theoretically free and proper (KFP) if the group trace tr of Gamma acts as the unit element in KKR Gamma (A, A). We show that free and proper Gamma-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if Gamma is torsion free and satisfies the KK Gamma-form of the Baum-Connes conjecture, then every Gamma-algebra satisfies (KFP). If alpha : Gamma -> U-n is a unitary representation and A satisfies property (KFP), we construct in a canonical way a rho class rho(A)(alpha) is an element of KKR/Z1,Gamma (A A) This construction generalizes the Atiyah-Patodi-Singer K-theory class with R/Z-coefficients associated to alpha. (C) 2015 Elsevier Inc. All rights reserved.
Author details: | Paolo Antonini, Sara AzzaliORCiD, Georges Skandalis |
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DOI: | https://doi.org/10.1016/j.jfa.2015.06.017 |
ISSN: | 0022-1236 |
ISSN: | 1096-0783 |
Title of parent work (English): | Journal of functional analysis |
Publisher: | Elsevier |
Place of publishing: | San Diego |
Publication type: | Article |
Language: | English |
Year of first publication: | 2016 |
Publication year: | 2016 |
Release date: | 2020/03/22 |
Tag: | Bivariant K-theory; Operator algebras; Rho invariants |
Volume: | 270 |
Number of pages: | 35 |
First page: | 447 |
Last Page: | 481 |
Funding institution: | European Research Council (E.R.C.) under European Union, ERC [291060]; University of Potsdam; [ANR-14-CE25-0012-01] |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
Peer review: | Referiert |