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 S_h be the subspace of self-adjoint elements. For any positive integer n, the n by n matrices with entries in S, denoted by M_n(S) is naturally a *-vector space.

For a concrete operator system X, there is a positive cone on each M_n(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, C_n of M_n(S)_h.

2. For each n, C_n∩ -C_n = {0}.

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

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 M_n and states on M_n(A). Given a c.p. map φ, the corresponding state f_φ is defined by f_φ(a_ij) = (1/n)Σ_ij φ(a_ij)_ij. The complete positivity of φ implies this is a positive functional. For the inverse, let f be a state on M_n(A). Define φ_f(a) = (f(a \otimes e_ij)).

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