Invariance results for pairings with algebraic K-theory

TL;DR

This paper introduces new finitely summable K-homology groups associated with complex algebras, which pair with cyclic homology and algebraic K-theory, extending the understanding of invariance in algebraic K-theory.

Contribution

It defines a new sequence of finitely summable K-homology groups and establishes their pairings with cyclic homology and algebraic K-theory, providing invariance results.

Findings

Pairing with cyclic homology via Chern-Connes character

Pairing with algebraic K-theory via Connes-Karoubi character

Introduction of finitely summable K-homology groups

Abstract

To each algebra over the complex numbers we associate a sequence of abelian groups in a contravariant functorial way. In degree (m-1) we have the m-summable Fredholm modules over the algebra modulo stable m-summable perturbations. These new finitely summable K-homology groups pair with cyclic homology and algebraic K-theory. In the case of cyclic homology the pairing is induced by the Chern-Connes character. The pairing between algebraic K-theory and finitely summable K-homology is induced by the Connes-Karoubi multiplicative character.