  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.
