The noncommutative Choquet boundary
Author:
William Arveson
Journal:
J. Amer. Math. Soc. 21 (2008), 1065-1084
MSC (2000):
Primary 46L07; Secondary 46L52
DOI:
https://doi.org/10.1090/S0894-0347-07-00570-X
Published electronically:
April 23, 2007
MathSciNet review:
2425180
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let be an operator system-a self-adjoint linear subspace of a unital
-algebra
such that
and
is generated by
. A boundary representation for
is an irreducible representation
of
on a Hilbert space with the property that
has a unique completely positive extension to
. The set
of all (unitary equivalence classes of) boundary representations is the noncommutative counterpart of the Choquet boundary of a function system
that separates points of
.
It is known that the closure of the Choquet boundary of a function system is the Šilov boundary of
relative to
. The corresponding noncommutative problem of whether every operator system has ``sufficiently many" boundary representations was formulated in 1969, but has remained unsolved despite progress on related issues. In particular, it was unknown if
for generic
. In this paper we show that every separable operator system has sufficiently many boundary representations. Our methods use separability in an essential way.
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Additional Information
William Arveson
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
Email:
arveson@math.berkeley.edu
DOI:
https://doi.org/10.1090/S0894-0347-07-00570-X
Keywords:
Choquet boundary,
operator system,
completely positive maps,
unique extension property
Received by editor(s):
January 12, 2007
Published electronically:
April 23, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.