Invariants

Abstract

In what follows we employ C*-homotopy theory to define invariants for C*-algebras. Section 5.1 introduces briefly bigraded homology and cohomology theories at large. The main examples are certain canonical extensions of KK-theory, see Section 5.2, and local cyclic homology theory, see Section 5.3, to the framework of pointed simplicial C*-spaces. We also observe that there is an enhanced Chern- Connes-Karoubi character between KK-theory and local cyclic theory on the level of simplicial C*-spectra. Section 5.5 deals with a form of K-theory of C*-algebras which is constructed using the model structures introduced earlier in this paper. This form of K-theory is wildly different from the traditional 2-periodic K-theory of C*-algebras [14, II] and relates to topics in geometric topology. Finally, in the last section we discuss zeta functions of C*-algebras.