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

TL;DR

This paper develops a bivariant K-theory framework for infinite dimensional spaces like SU(∞), extending twisted K-theory and K-homology, and introduces a Chern-Connes type character linking these theories to cyclic homology.

Contribution

It constructs a new bivariant K-theory for separable σ-C^*-algebras that unifies twisted K-theory and K-homology, and defines a bivariant Chern-Connes character.

Findings

Established a bivariant K-theory for infinite dimensional spaces

Defined a bivariant Chern-Connes type character to cyclic homology

Analyzed the dual Chern-Connes character under certain hypotheses

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^*-algebras that generalizes both twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern--Connes type character taking values in bivariant local cyclic homology. We analyse 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^*-algebras (in the compact case).