Boundaries for Operator Systems

Let H be a complex Hilbert space, B(H) the bounded linear operators on H, and S a unital, linear subspace of B(H). The set S is an operator system. We investigate boundaries, or norm-attaining sets, for operator systems. In particular, we show that Arveson's noncommutative Choquet boundary for a separable system forms a boundary. We obtain a new characterization of unital completely positive maps having the unique extension property. We show the relationship between various sorts of extreme points of noncommutative convex sets, relate them to boundary representations for operator systems, and prove that operator systems in matrix algebras have a minimal boundary. Finally, we explore noncommutative peaking phenomena for operator systems, proving a partial generalization of the Bishop-de Leeuw theorem for uniform algebras.

Note: Abstract extracted from PDF text