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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
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

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.

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