Let S be an operator system – a self-adjoint linear subspace 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 representation π of C∗(S) on a Hilbert space with the property that π S has a unique completely positive extension to C∗(S). The set ∂S of all (unitary equivalence classes of) boundary representations is the nonc...

An axiomatization of the concept of entropy of a discrete Choquet capacity is given. It is based on three axioms: the symmetry property, a boundary condition for which the entropy reduces to the classical Shannon entropy, and a generalized version of the well-known recursivity property. This entropy, recently introduced to extend the Shannon entropy to nonadditive measures, fulfills several pro...

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 complet...

A (finite or countably infinite) set G of generators of an abstract C∗-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space A ⊆ B(H) and every sequence of unital completely positive linear maps φ1, φ2, . . . from B(H) to itself, lim n→∞ ‖φn(g)− g‖ = 0,∀g ∈ G =⇒ lim n→∞ ‖φn(a)− a‖ = 0, ∀a ∈ A. We show that one can determine whether a given set G of generato...

Let L be a closed subspace of C(X) which separates points and contains the constants. Denote the Korovkin closure of L by L̂. Then L = L̂ ∩ ML where ML = {f ∈ C(X) : ∫ f d(μ ◦ j) = 0 for all boundary dependences μ on KL}. We consider the relation between L and ML, the Choquet boundary of ML and the state space of ML. 1980 AMS Mathematics Subject Classification (1985 Revision): Primary 47B38, 47B5...