Abstract: Subalgebras of C*-algebras III,
Multivariable operator theory
by William Arveson
A d-contraction is a d-tuple (T_1,\dots,T_d)
of mutually commuting operators acting on a common
Hilbert space H such that
$$
\|T_1\xi_1+T_2\xi_2+\dots +T_d\xi_d\|^2\leq
\|\xi_1\|^2+\|\xi_2\|^2+\dots+\|\xi_d\|^2
$$
for all $\xi_1,\xi_2,\dots,\xi_d\in H$.
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.
NOTE: this paper has appeared in Acta Mathematica,
vol 181 (1998), pp. 159--228.