The Choquet boundary of an operator system

Abstract

Abstract. We show that every operator system (and hence every unital operator algebra) has sufficiently many boundary represen-tations to generate the C*-envelope. We solve a 45 year old problem of William Arveson that is central to his approach to non-commutative dilation theory. We show that every operator system and every unital operator algebra has sufficiently many boundary representations to completely norm it. Thus the C*-algebra generated by the image of the direct sum of these maps is the C*-envelope. This was a central problem left open in Arveson’s seminal work [2] on dilation theory for arbitrary operator algebras. In the intervening years, the existence of the C*-envelope was established, but a general argument producing boundary representations has not been available. Arveson [2, 3] reformulated the classical dilation theory of Sz. Nagy [14] so that it made sense for an arbitrary unital closed subalgebra A of a C*-algebra. A central theme was the use of completely pos-itive and completely bounded maps. He proposed the existence of a family of special representations of A, called boundary representations, which have unique completely positive extensions to C∗(A) that are irreducible ∗-representations. The set of boundary representations is a noncommutative analogue of the Choquet boundary of a function alge-bra, i.e. the set of points with unique representing measures. Arveson proposed that there should be sufficiently many boundary representa-tions, so that their direct sum recovers the norm on Mn(A) for all n ≥ 1. In this case, he showed that the C*-algebra generated by this direct sum enjoys an important universal property, and provides a re-alization of the C*-envelope of A.