 

Ilan Barnea, Michael Joachim, and Snigdhayan Mahanta
Model structure on projective systems of C*algebras and bivariant homology theories view print


Published: 
March 19, 2017 
Keywords: 
Procategory, model
category,
∞category, triangulated category, bivariant homology, KKtheory, 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
procategory 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 wellknown category of fibrant objects structure on SC*. We show that the procategory Pro(SC*) also contains, as a full coreflective subcategory, the category of proC*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 ∞categorical counterparts were constructed earlier by the third author. Finally, we use our model structure to develop a bivariant Ktheory 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.unimuenster.de
Snigdhayan Mahanta:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
snigdhayan@gmail.com

