Subalgebras of C∗-algebras Iii: Multivariable Operator Theory

A d-contraction is a d-tuple (T1, . . . , Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · ·+ Tdξd‖ 2 ≤ ‖ξ1‖ 2 + ‖ξ2‖ 2 + · · ·+ ‖ξd‖ 2 for all ξ1, ξ2, . . . , ξd ∈ H. These are the higher dimensional counterparts of contractions. We show that many of the operator-theoretic aspects of function theory in the unit disk generalize to the unit ball Bd in complex d-space, including von Neumann’s inequality and the model theory of contractions. These results depend on properties of the d-shift, a distinguished d-contraction which acts on a new H space associated with Bd, and which is the higher dimensional counterpart of the unilateral shift. H 2 and the d-shift are highly unique. Indeed, by exploiting the noncommutative Choquet boundary of the d-shift relative to its generated C∗-algebra we find that there is more uniqueness in dimension d ≥ 2 than there is in dimension one.