Finite and torsion KK-theories

TL;DR

This paper introduces finite and torsion KK-theories for C*-algebras, extending algebraic K-theory results, and explores their properties, relationships, and implications for the Baum-Connes Conjecture.

Contribution

It develops finite and torsion KK-theories, extending algebraic K-theory results, and formulates finite, torsion, and rational Baum-Connes Conjecture variants.

Findings

Extended Browder-Karoubi-Lambre's theorem to finite KKG-theory.

Defined torsion KK-theory as a direct limit of finite KK-theories.

Formulated finite, torsion, and rational Baum-Connes Conjecture versions.

Abstract

We develop a finite KKG-theory of C*-algebras following Arlettaz- H.Inassaridze's approach to finite algebraic K-theory. The Browder- Karoubi-Lambre's theorem on the orders of the elements for finite algebraic K-theory is extended to finite KKG-theory. A new bivariant theory, called torsion KK-theory is defined as the direct limit of finite KK-theories. Such bivariant K-theory has almost all KKG-theory properties and one has a long exact sequence relating KK-theory, rational bivariant K-theory and torsion KK-theory. For a given homology theory on the category of separable GC*-algebras finite, rational and torsion homology theories are introduced and investigated. In particular, we formulate finite, torsion and rational versions of Baum-Connes Conjecture. The later is equivalent to the investigation of rational and q-finite analogues for Baum-Connes Conjecture for all prime q.