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
ξ
1
+
T
2
ξ
2
+...+
T
d
ξ
d
∥
2
≤∥
ξ
1
∥
2
+∥
ξ
2
∥
2
+...+∥
ξ
d
∥
2
for all
ξ
1
,
ξ
2
,...,
ξ
d
∈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≥2
than there is in dimension one.