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

摘要

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 ∞-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.