 
Summary: THE NONCOMMUTATIVE CHOQUET BOUNDARY
WILLIAM ARVESON
Abstract. Let S be an operator system a selfadjoint linear sub
space of a unital C
algebra A such that 1 S and A = C
(S) is
generated by S. A boundary representation for S is an irreducible rep
resentation of C
(S) on a Hilbert space with the property that S
has a unique completely positive extension to C
(S). The set S of
all (unitary equivalence classes of) boundary representations is the non
commutative counterpart of the Choquet boundary of a function system
S C(X) that separates points of X.
It is known that the closure of the Choquet boundary of a function
system S is the Silov boundary of X relative to S. The corresponding
noncommutative problem of whether every operator system has "suf
ficiently many" boundary representations was formulated in 1969, but
has remained unsolved despite progress on related issues. In particu
lar, it was unknown if S = for generic S. In this paper we show
