Noncommutative Differential K-theory

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. Noncommutative differentialK-theory 37 4.1. Noncommutative differential K-theory with Karoubi's Chern character 37 4.2. Cycle maps 41 4.3. Hexagon diagram 44 Appendix A. The Serre-Swan theorem and its differential refinements 48