Smooth functions on algebraic K-theory

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.