D ec 1 99 7 PARTIAL DYNAMICAL SYSTEMS AND C ∗ - ALGEBRAS GENERATED BY PARTIAL ISOMETRIES

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 Calgebra 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.