  The purpose of this work is to give a definition of a topological K-theory
for dg-categories over C and to prove that the Chern character map from
algebraic K-theory to periodic cyclic homology descends naturally to this new
invariant. This topological Chern map provides a natural candidate for the
existence of a rational structure on the periodic cylic homology of a smooth
proper dg-algebra, within the theory of noncommutative Hodge structures. The
definition of topological K-theory consists in two steps : taking the
topological realization of algebraic K-theory, and inverting the Bott element.
The topological realization is the left Kan extension of the functor "space of
complex points" to all simplicial presheaves over complex algebraic varieties.
Our first main result states that the topological K-theory of the unit
dg-category is the spectrum BU. For this we are led to prove a homotopical
generalization of Deligne's cohomological proper descent, using Lurie's proper
descent. The fact that the Chern character descends to topological K-theory is
established by using Kassel's K\"unneth formula for periodic cyclic homology
and once again the proper descent result. In the case of a dg-category of
perfect complexes on a smooth scheme, we show that we recover the usual
topological K-theory. Finally in the case of a finite dimensional associative
algebra, we show that the lattice conjecture holds. This gives a formula for
the periodic homology groups of a finite dimensional algebra in terms of the
stack of projective modules of finite type.
