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Journal of the American Mathematical Society

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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
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Abstract: Let $S$ be an operator system–a self-adjoint linear subspace of a unital $C^*$-algebra $A$ such that $\mathbf 1\in S$ and $A=C^*(S)$ is generated by $S$. A boundary representation for $S$ is an irreducible representation $\pi$ of $C^*(S)$ on a Hilbert space with the property that $\pi \restriction _S$ has a unique completely positive extension to $C^*(S)$. The set $\partial _S$ of all (unitary equivalence classes of) boundary representations is the noncommutative counterpart of the Choquet boundary of a function system $S\subseteq C(X)$ that separates points of $X$. It is known that the closure of the Choquet boundary of a function system $S$ is the Šilov boundary of $X$ relative to $S$. 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 $\partial _S\neq \emptyset$ for generic $S$. 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

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.