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.