Noncommutative Differential K-theory

TL;DR

This paper develops a noncommutative differential K-theory framework using Karoubi's Chern character, extending classical differential K-theory to noncommutative algebras with a new secondary transgression theory.

Contribution

It introduces a differential extension of algebraic K-theory for noncommutative algebras, including a differential refinement of the Serre--Swan correspondence and a noncommutative cohomology hexagon.

Findings

Recovers differential K-theory for smooth manifolds

Establishes a noncommutative differential cohomology hexagon

Develops secondary transgression forms for noncommutative algebras

Abstract

We introduce a differential extension of algebraic K-theory of an algebra using Karoubi's Chern character. In doing so, we develop a necessary theory of secondary transgression forms as well as a differential refinement of the smooth Serre--Swan correspondence. Our construction subsumes the differential K-theory of a smooth manifold when the algebra is complex-valued smooth functions. Furthermore, our construction fits into a noncommutative differential cohomology hexagon diagram.