The Choquet Boundary of an Operator System

We show that every operator system (and hence every unital operator algebra) has sufficiently many boundary representations 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 positive 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 algebra, i.e. the set of points with unique representing measures. Arveson proposed that there should be sufficiently many boundary representations, 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 realization of the C*-envelope of A. 2010 Mathematics Subject Classification. 46L07, 46L52, 47A20, 47L55.