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Summary: THE NONCOMMUTATIVE CHOQUET BOUNDARY
WILLIAM ARVESON
Abstract. Let S be an operator system a self-adjoint 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
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