Representation and index theory for C^*-algebras generated by commuting isometries

We discuss here representation and Fredholm theory for C^*-algebras generated by commuting isometries. More particularly, for n commuting isometries {V_j: 1 ≤ j ≤ n} on separable Hilbert space we give a representation resembling the well-known representation for a single isometry. Our representation permits an analysis of the C^*-algebras Ol=Ol(V_j:1≤j≤n) generated by the {V_j}. The commutator ideal in Ol is identified precisely and, under certain additional hypotheses, the Fredholm operators in Ol are also precisely determined. Finally, we obtain formulas in terms of topological data for the index of Fredholm operators in some interesting algebras of the type Ol(V_j:1≤j≤n).