Twisted K-theory, K-homology, and bivariant Chern–Connes type character of some infinite dimensional spaces

Abstract

We study the twisted K-theory and K-homology of some infinite dimensional spaces, like $SU(\infty )$, in the bivariant setting. Using a general procedure due to Cuntz we construct a bivariant K-theory on the category of separable $\sigma$-$C^{\ast }$-algebras that generalizes both the twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern–Connes-type character taking values in a bivariant local cyclic homology. We analyze the structure of the dual Chern–Connes character from (analytic) K-homology to local cyclic cohomology under some reasonable hypotheses. We also investigate the twisted periodic cyclic homology via locally convex algebras and the local cyclic homology via $C^{\ast }$-algebras (in the compact case).