Unbounded bivariant $K$-theory and correspondences in noncommutative geometry

TL;DR

This paper develops a new algebraic framework for unbounded bivariant K-theory using smooth connections on KK-cycles, enabling direct Kasparov product computation within noncommutative geometry.

Contribution

It introduces a notion of smooth connection for unbounded KK-cycles and a framework of smooth algebras, facilitating algebraic Kasparov product definition.

Findings

Defined smooth connections for unbounded KK-cycles

Established a framework of smooth algebras and differentiable modules

Reformulated KK-cycles as morphisms in a spectral triples category

Abstract

By introducing a notion of smooth connection for unbounded $KK$-cycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable $C^{*}$-module. The theory of operator spaces provides the required tools. Finally, the above mentioned $KK$-cycles with connection can be viewed as the morphisms in a category whose objects are spectral triples.