  In the first part of the article we introduce $C^*$-algebras associated to
self-similar groups and study their properties and relations to known algebras.
The algebras are constructed as sub-algebras of the Cuntz-Pimsner algebra (and
its homomorphic images) associated with the self-similarity of the group. We
study such properties as nuclearity, simplicity and Morita equivalence with
algebras related to solenoids.
  The second part deals with the Schur complement transformations of elements
of self-similar algebras. We study properties of such transformations and apply
them to the spectral problem for Markov type elements in self-similar
$C^*-$algebras. This is related to the spectral problem of the discrete Laplace
operator on groups and graphs. Application of the Schur complement method in
many situations reduces the spectral problem to study of invariant sets (very
often of the type of a ``strange attractor'') of a multidimensional rational
transformation. A number of illustrating examples is provided. Finally we
observe a relation between the Schur complement transformations and
Bartholdi-Kaimanovich-Virag transformations of random walks on self-similar
groups.
