The noncommutative Choquet boundary

By W. Arveson. 2007

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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- resentationof 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 y 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 6 ∅ 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3eb897a47e7230940dbe387fdc1b812b41220a60

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Citations in the corpus

The Choquet boundary of an operator system (2013, methodology): “Arveson [4] revisited the problem of the existence of boundary representations using the ideas of Dritschel and McCullough.”
Semicrossed Products of Operator Algebras by Semigroups (2014, background): “Arveson [9] showed that in the separable case, A always has enough boundary representations so that their direct sum is completely isometric.”
Dilation theory, commutant lifting and semicrossed products (2011, background): “Arveson [10] provides a refinement of this result in the separable case by showing that there are sufficiently many irreducible maximal representations, which are the boundary representations that Arveson introduced in [5] as an analogue of the Choquet boundary of a function algebra.”
The noncommutative Choquet boundary II: Hyperrigidity (2008, background): “In a previous paper [ Arv08 ] it was shown that every separable operator system has sufficiently many boundary representations, thereby providing a noncommutative counterpart of the function-theoretic fact that the closure of the Choquet boundary is the Silov boundary.”
Operator Algebras for Multivariable Dynamics (2007, background): “ctive. Dritschel and McCullough [12] showed that the direct sum of all (possibly reducible) boundary representations yields a faithful representation of the C*-envelope, C∗(A). Very recently, Arveson [4] has shown that these reducible boundary representations are, in fact, direct integrals of repres”