User:Mct mht/Choi-Effros for operator systems

Concrete operator systems

An operator system is a vector subspace of a C*-algebra that is also closed under the *-operation.

Abstract characterization of operator systems A C*-algebra can be abstractly characterized as a Banach algebra equipped with a *-operation that satisfies the C*-identity: ||x*x|| = ||x||2 for every element x. By the GNS construction and the abundance of states, every abstract C*-algebra is isomorphic to a norm-closed, *-closed subalgebra of B(H), for some Hilbert space H.

Operator systems can be similarly characterized, as vector spaces with a *-operation, and a matrix order with suitable properties.

By a *-vector space, we simply mean a vector space with an adjoint operation *. Let S be a *-vector space and Sh be the subspace of self-adjoint elements. For any positive integer n, the n by n matrices with entries in S, denoted by Mn(S) is naturally a *-vector space.

For a concrete operator system X, there is a positive cone on each Mn(X) consisting of elements that, when viewed in the ambient C*-algebra, are positive. This property needs to be abstracted. A *-vector space is said to have a matrix order if


 * 1. For each n, there exists a distinguished subset, called a positive cone, Cn of Mn(S)h.


 * 2. For each n, Cn∩ -Cn = {0}.


 * 3. For all m by n scalar matrices a and x in Cn, a x a* lies in Cm.⊉

Finally, we consider the role of the unit. An element e in S is said to be an order unit if

Archimedean order unit

Order units are necessarily positive

Lemma Order-induced norm on matrices. Lemma Correspondence between states and c.p. maps

There exist an one to one correspondence between c.p. maps from A to Mn and states on Mn(A). Given a c.p. map φ, the corresponding state fφ is defined by fφ(aij) = (1/n)Σij φ(aij)ij. The complete positivity of φ implies this is a positive functional. For the inverse, let f be a state on Mn(A). Define φf(a) = (f(a \otimes eij)).

Statement of the theorem

Any *-vector space with an Archimedean matrix order and a distinguised order unit e is isomorphic to a concrete operator system.

Outline of proof