Characterization

Choi and Effros [CE77, Theorem 4.4] showed the following characterization theorem:

Let $V$ be an operator system. Then there are a Hilbert space $\H$ and a unital complete order isomorphism from $V$ to a selfadjoint subspace of $B(\H)$.

A unital complete order isomorphism is obtained by

$$\Phi:V \rightarrow \bigoplus_{n\in{\mathbb{N}}}\bigoplus_{\varphi\in S_n} {\mathbb{M}}_n$$

$$x \mapsto (\varphi(x))_\varphi,$$

where $S_n$ is the set of all unital completely positive maps $\varphi:V\rightarrow {\mathbb{M}}_n$.