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Transactions of the American Mathematical Society

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The KK-theory of fundamental C*-algebras


Authors: Pierre Fima and Emmanuel Germain
Journal: Trans. Amer. Math. Soc. 370 (2018), 7051-7079
MSC (2010): Primary 19K35, 46L05, 46L80
DOI: https://doi.org/10.1090/tran/7211
Published electronically: June 26, 2018
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Abstract: Given a graph of C*-algebras as defined in [Adv. Math. 260 (2014), 233-280], we prove a long exact sequence in KK-theory similar to the one obtained by Pimsner in [Invent. Math. 86 (1986), 603-634] for both the maximal and the vertex-reduced fundamental C*-algebras of the graph in the presence of possibly non-GNS-faithful conditional expectations. We deduce from it the KK-equivalence between the full fundamental C*-algebra and the vertex-reduced fundamental C*-algebra even for non-GNS-faithful conditional expectations. Our results unify, simplify, and generalize all the previous results obtained by Cuntz, Pimsner, Germain, and Thomsen. They also generalize the previous results of the authors on amalgamated free products.


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Additional Information

Pierre Fima
Affiliation: Université Paris Diderot, Sorbonne Paris Cité, IMJ-PRG, UMR 7586, F-75013, Paris, France – and – Sorbonne Universités, UPMC Paris 06, UMR 7586, IMJ-PRG, F-75005, Paris, France – and – CNRS, UMR 7586, IMJ-PRG, F-75005, Paris, France
Email: pierre.fima@imj-prg.fr

Emmanuel Germain
Affiliation: LMNO, CNRS UMR 6139, Université de Caen, 14032 Caen, France
Email: emmanuel.germain@unicaen.fr

DOI: https://doi.org/10.1090/tran/7211
Received by editor(s): March 4, 2016
Received by editor(s) in revised form: January 23, 2017, and February 19, 2017
Published electronically: June 26, 2018
Additional Notes: The first author was partially supported by ANR grants OSQPI and NEUMANN
The second author thanks CMI, Chennai for its support when part of this research was under way.
Article copyright: © Copyright 2018 American Mathematical Society

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