Topological methods for<i>C</i>^∗-algebras. III. Axiomatic homology

Abstract

A homology theory consists of a sequence {h n } of covariant functors from a suitable category of C*-algebras to abelian groups which satisfies homotopy and exactness axioms. We show that such theories have Mayer-Vietoris sequences and (if additive) commute with inductive limits. There are analogous definitions and theorems in cohomology with one important difference: an additive cohomology theory associates a Milnor lim 1 sequence to an inductive limit of C*-aIgebras. As prerequisite to these results we develop the necessary homotopy theory, including cofibrations and cofibre theories.