Subalgebras of C*-algebras III: multivariable operator theory

Abstract

A d-contraction is a d-tuple $(T_1,...,T_d)$ of mutually commuting operators acting on a common Hilbert space H such that $ \|T_1ξ_1+T_2ξ_2+... +T_dξ_d\|^2\leq \|ξ_1\|^2+\|ξ_2\|^2+...+\|ξ_d\|^2 $ for all $ξ_1,ξ_2,...,ξ_d\in H$. 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 B_d 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 $H^2$ space associated with B_d, 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 $d\geq 2$ than there is in dimension one.