  A collection of partial isometries whose range and initial projections
satisfy a specified set of conditions often gives rise to a partial
representation of a group. The C*-algebra generated by the partial isometries
is thus a quotient of the universal C*-algebra for partial representations of
the group, from which it inherits a crossed product structure, of an abelian
C*-algebra by a partial action of the group. Questions of faithfulness of
representations, simplicity, and ideal structure of these C*-algebras can then
be addressed in a unified manner from within the theory of partial actions. We
do this here, focusing on two key properties of partial dynamical systems,
namely amenability and topological freeness; they are the essential ingredients
of our main results in which we characterize faithful representations,
simplicity and the ideal structure of crossed products. As applications we
consider three situations involving C*-algebras generated by partial
isometries: partial representations of groups, Toeplitz algebras of
quasi-lattice ordered groups, and Cuntz-Krieger algebras. These C*-algebras
share a crossed product structure which we give here explicitly and which we
use to study them in terms of the underlying partial actions.
