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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The KK-theory of fundamental C*-algebras
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by Pierre Fima and Emmanuel Germain PDF
Trans. Amer. Math. Soc. 370 (2018), 7051-7079 Request permission

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
  • 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.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 7051-7079
  • MSC (2010): Primary 19K35, 46L05, 46L80
  • DOI: https://doi.org/10.1090/tran/7211
  • MathSciNet review: 3841842