21. Homology and Cohomology Theories on C*-Algebras

In this section, we define and derive some basic properties of abstract homology and cohomology theories for C*-algebras. We consider only trivially graded C*-algebras.

The results and ideas of this section are taken primarily from [Schochet 1984a], although many of them are at least based on folklore and some are straightforward translations of standard results of topology.

21.1. Basic Definitions

Let S be a subcategory of the category of all C*-algebras, which is closed under quotients, extensions, and closed under suspension in the sense that if A ∈ Ob(S),

then SA ∈ Ob((S), that ∅ ∈ Homs(A,B) implies S∅ ∈ Homs(SA,SB), and that C0(ℝ) ∈ Ob(S) and every *-homomorphism from C0(ℝ) to A is in Horns (C0 (ℝ), A).

We will consider covariant and contravariant functors F from (S to Ab, the category of abelian groups, although we could more generally work with functors to any abelian category with arbitrary limits (for some results less is required). We will consider only functors which satisfy the following homotopy axiom.