Commutant lifting and interpolation on the unit ball

We solve the commutant lifting and interpolation problems in the setting of the Hardy space and Schur functions on the open unit ball of ${\mathbb{C}}^n$. Our solutions also signify the role of inner functions on the unit ball, objects whose existence was once in doubt and to which Aleksandrov, Rudin, and others made fundamental contributions. Some of our results, particularly those related to inner functions, are new even in the classical, one-variable case.