  We present a systematic study of the structure of crossed products and fixed
point algebras by compact group actions with the Rokhlin property on not
necessarily unital C*-algebras. Our main technical result is the existence of
an approximate homomorphism from the algebra to its subalgebra of fixed points,
which is a left inverse for the canonical inclusion. Upon combining this with
results regarding local approximations, we show that a number of classes
characterized by inductive limit decompositions with weakly semiprojective
building blocks, are closed under formation of crossed products by such
actions. Similarly, in the presence of the Rokhlin property, if the algebra has
any of the following properties, then so do the crossed product and the fixed
point algebra: being a Kirchberg algebra, being simple and having tracial rank
zero or one, having real rank zero, having stable rank one, absorbing a
strongly self-absorbing C*-algebra, satisfying the Universal Coefficient
Theorem (in the simple, nuclear case), and being weakly semiprojective. The
ideal structure of crossed products and fixed point algebras by Rokhlin actions
is also studied. The methods of this paper unify, under a single conceptual
approach, the work of a number of authors, who used rather different
techniques. Our methods yield new results even in the well-studied case of
finite group actions with the Rokhlin property.
