  We introduce a method to study C*-algebras possessing an action of the circle
group, from the point of view of its internal structure and its K-theory. Under
relatively mild conditions our structure Theorem shows that any C*-algebra,
where an action of the circle is given, arises as the result of a construction
that generalizes crossed products by the group of integers.
  Such a generalized crossed product construction is carried out for any
partial automorphism of a C*-algebra, where by a partial automorphism we mean
an isomorphism between two ideals of the given algebra.
  Our second main result is an extension to crossed products by partial
automorphisms, of the celebrated Pimsner-Voiculescu exact sequence for
K-groups.
  The representation theory of the algebra arising from our construction is
shown to parallel the representation theory for C*-dynamical systems. In
particular, we generalize several of the main results relating to regular and
covariant representations of crossed products.
