New York Journal of Mathematics
Volume 23 (2017) 383-439

  

Ilan Barnea, Michael Joachim, and Snigdhayan Mahanta

Model structure on projective systems of C*-algebras and bivariant homology theories

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Published: March 19, 2017
Keywords: Pro-category, model category, ∞-category, triangulated category, bivariant homology, KK-theory, C*-algebra
Subject: 46L85, 46L80, 18E30, 18G55, 55P42, 55N15

Abstract
Using the machinery of weak fibration categories due to Schlank and the first author, we construct a convenient model structure on the pro-category of separable C*-algebras Pro(SC*). The opposite of this model category models the ∞-category of pointed noncommutative spaces NS defined by the third author. Our model structure on Pro(SC*) extends the well-known category of fibrant objects structure on SC*. We show that the pro-category Pro(SC*) also contains, as a full coreflective subcategory, the category of pro-C*-algebras that are cofiltered limits of separable C*-algebras. By stabilizing our model category we produce a general model categorical formalism for triangulated and bivariant homology theories of C*-algebras (or, more generally, that of pointed noncommutative spaces), whose stable ∈fty-categorical counterparts were constructed earlier by the third author. Finally, we use our model structure to develop a bivariant K-theory for all projective systems of separable C*-algebras generalizing the construction of Bonkat and show that our theory naturally agrees with that of Bonkat under some reasonable assumptions.

Acknowledgements

The first author was supported by the Alexander von Humboldt Foundation (Humboldt Professorship of Michael Weiss).
The third author was supported by the Deutsche Forschungsgemeinschaft (SFB 878 and SFB 1085), ERC through AdG 267079, and the Alexander von Humboldt Foundation (Humboldt Professorship of Michael Weiss).


Author information

Ilan Barnea:
Department of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem, 9190401, Israel
ilanbarnea770@gmail.com

Michael Joachim:
Mathematical Institute, University of Muenster, Einsteinstrasse 62, 48149 Muenster, Germany
joachim@math.uni-muenster.de

Snigdhayan Mahanta:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
snigdhayan@gmail.com