Subalgebras of C*-algebras III: multivariable operator theory

TL;DR

This paper extends classical operator theory concepts from the unit disk to higher dimensions, exploring properties of d-contractions, the d-shift, and their associated function spaces, revealing increased uniqueness in higher dimensions.

Contribution

It generalizes key operator-theoretic results to the unit ball in complex d-space, introducing the d-shift and analyzing its unique properties in higher dimensions.

Findings

von Neumann's inequality holds in higher dimensions

The d-shift acts on a new H^2 space associated with B_d

Greater uniqueness of the d-shift in dimensions d ≥ 2

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_1\xi_1+T_2\xi_2+... +T_d\xi_d\|^2\leq \|\xi_1\|^2+\|\xi_2\|^2+...+\|\xi_d\|^2 $ for all $\xi_1,\xi_2,...,\xi_d\in H$. 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.