Operator system

Abstract: 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.

Abstract: Given an Archimedean order unit space (V, V , e), we construct a minimal operator system OMIN(V ) and a maximal operator system OMAX(V ), which are the analogues of the minimal and maximal operator spaces of a normed space. We develop some of the key properties of these operator systems and make some progress on characterizing when an operator system S is completely boundedly isomorphic to either OMIN(S) or to OMAX(S). We then apply these concepts to the study of entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN(Mn) to OMAX(Mm) if and only if it is entanglement breaking.

Abstract: Let S be an operator system -- a self-adjoint linear subspace of a unital C*-algebra A such that contains 1 and A=C*(S) is generated by S. A boundary representation for S is an irreducible representation \pi of C*(S) on a Hilbert space with the property that $\pi\restriction_S$ has a unique completely positive extension to C*(S). The set $\partial_S$ of all (unitary equivalence classes of) boundary representations is the noncommutative counterpart of the Choquet boundary of a function system $S\subseteq C(X)$ that separates points of X. It is known that the closure of the Choquet boundary of a function system S is the Silov boundary of X relative to S. The corresponding noncommutative problem of whether every operator system has "sufficiently many" boundary representations was formulated in 1969, but has remained unsolved despite progress on related issues. In particular, it was unknown if $\partial_S$ is nonempty 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.

Abstract: We show that every operator system (and hence every unital operator algebra) has sufficiently many boundary representations to generate the C*-envelope.