  In this paper we study the problem of approximation of the $L^2$-topological
invariants by their finite dimensional analogues. We obtain generalizations of
the theorem of L\"uck, dealing with towers of finitely sheeted normal
coverings. We prove approximation theorems, establishing relations between the
homological invariants, corresponding to infinite dimensional representations
and sequences of finite dimensional representations, assuming that their
normalized characters converge. Also, we find an approximation theorem for
residually finite $p$-groups ($p$ is a prime), where we use the homology with
coefficients in a finite field $\fp$.
  We view sequences of finite dimensional flat bundles of growing dimension as
examples of growth processes. We study a von Neumann category with a Dixmier
type trace, which allows to describe the asymptotic invariants of growth
processes. We introduce a new invariant of torsion objects, the torsion
dimension. We show that the torsion dimension appears in general as an
additional correcting term in the approximation theorems; it vanishes under
some arithmeticity assumptions. We also show that the torsion dimension allows
to establish non-triviality of the Grothendieck group of torsion objects.
