  The pro-algebraic fundamental group can be understood as a completion with
respect to finite-dimensional non-commutative algebras. We introduce finer
invariants by looking at completions with respect to Banach and C*-algebras,
from which we can recover analytic and topological representation spaces,
respectively. For a compact Kaehler manifold, the C*-completion also gives the
natural setting for non-abelian Hodge theory; it has a pure Hodge structure, in
the form of a pro-C*-dynamical system. Its representations are pluriharmonic
local systems in Hilbert spaces, and we study their cohomology, giving a
principle of two types, and splittings of the Hodge and twistor structures.
