  For any complex scheme X or any dg category, there is an associated K-theory
presheaf on the category of complex affine schemes. We study real smooth
functions on this presheaf, defined by Kan extension, and show that they are
closely related to real Deligne cohomology. When X is quasi-compact and
semi-separated, and for various non-commutative derived schemes, these smooth
functions on K-theory are dual to the homotopy fibre of the Chern character
from Blanc's semi-topological K-theory to cyclic homology.
