Definitions

Let $V$ be a complex vector space. An involution on $V$ is a conjugate linear map $*:V\rightarrow V$, $v\mapsto v^*$, such that $v^{**}=v$. A complex vector space is an involutive vector space if there is an involution on $V$. Let $V$ be an involutive vector space. Then $V_\mathrm{sa}$ is the real vector space of selfadjoint elements of $V$, i.e. those elements of $V$, such that $v^*=v$. An involutive vector space is an ordered vector space if there is a proper cone²⁰ $V^+\subset V_{\mathrm{sa}}$. The elements of $V^+$ are called positive and there is an order on $V_\mathrm{sa}$ defined by $v\leq w$ if $w-v\in V^+$ for $v,w\in V_{\mathrm{sa}}$.

An element $\mathrm{1\!\!\!\:l}\in V^+$ is an order unit if for any $v\in V_{\mathrm{sa}}$ there is a real number $t>0$, such that $-t\mathrm{1\!\!\!\:l}\leq v\leq t\mathrm{1\!\!\!\:l}$. If $V_+$ has an order unit then $V_{\mathrm{sa}}=V_+-V_+$.

The cone $V^+$ is called Archimedian if $w\in -V^+$ whenever there exists $v\in V_{\mathrm{sa}}$ such that $tw\leq v$ for all $t>0$. If there is an order unit $\mathrm{1\!\!\!\:l}$ then $V_+$ is Archimedian if $w\in -V^+$ whenever $tw\leq \mathrm{1\!\!\!\:l}$ for all $t>0$.

Let $V$ be an ordered vector space. If $V_+$ is Archimedian and contains a distinguished order unit $\mathrm{1\!\!\!\:l}$ then $(V,\mathrm{1\!\!\!\:l})$ is called an ordered unit space.

Let $V,W$ be involutive vector spaces. We define an involution $\star$ on the space $L(V,W)$ of all linear mappings from $V\rightarrow W$ by $\varphi^\star(v)=\varphi(v^*)^*$, $\varphi\in L(V,W)$. If moreover $V,W$ are ordered vector spaces, then $\varphi$ is positive if $\varphi^\star=\varphi$ and $\varphi(V^+)\subset W^+$. If $(V,\mathrm{1\!\!\!\:l})$ and $(W,\mathrm{1\!\!\!\:l}')$ are ordered unit spaces a positve map $\varphi:V\rightarrow W$ is called unital if $\varphi(\mathrm{1\!\!\!\:l})=\mathrm{1\!\!\!\:l}'$.

Let $V$ be an involutive vector space. Then $M_n(V)$ is also an involutive vector space by $[v_{ij}]^*=[v_{ji}^*]$. $V$ is a matrix ordered vector space if there are proper cones $M_n(V)^+\subset M_n(V)_{\mathrm{sa}}$ for all $n\in{\mathbb{N}}$, such that $\alpha^*M_p(V)^+\alpha\subset M_q(V)^+$ for all $\alpha\in M_{pq}$ and $p,q\in{\mathbb{N}}$ holds²¹. This means that $(M_n(V)_+)_{n\in{\mathbb{N}}}$ is a matrix cone .

Let $V,W$ be matrix ordered vector spaces. A linear mapping $\phi:V\rightarrow W$ is completely positive if $\phi^{(n)}:M_n(V)\rightarrow M_n(W)$ is positive for all $n\in{\mathbb{N}}$. A complete order isomorphism²² from $V$ to $W$ is a completely positive map from $V\rightarrow W$ that is bijective, such that the inverse map is completely positive.

The well-known Stinespring theorem for completely positive maps reads [Pau86, Theorem 4.1]:

Let $A$ be a unital $C^*$-algebra and let $\H$ be a Hilbert space. If $\psi:A\rightarrow B(H)$ is completely positive then there are a Hilbert space $H_\pi$, a unital $*$-homomorphism $\pi:A\rightarrow B(H_\pi)$ and a linear mapping $V:H\rightarrow H_\pi$, such that $\psi(a)=V^*\pi(a)V$ for all $a\in A$.

Let $V$ be an involutive vector space. Then $V$ is called an operator system if it is a matrix ordered ordered unit space, such that $M_n(V)^+$ is Archimedian for all $n\in{\mathbb{N}}$. In this case $M_n(V)$ is an ordered unit space with order unit $\mathrm{1\!\!\!\:l}_n=\mathrm{1\!\!\!\:l}\otimes{\mathrm{id}}_{M_n}$ for all $n\in{\mathbb{N}}$, where $\mathrm{1\!\!\!\:l}\in V_+$ is the distinguished order unit of $V$ and ${\mathrm{id}}_{M_n}$ is the unit of ${\mathbb{M}}_n$.

Example

Let $\H$ be a Hilbert space. Then, obviously, $B(\H)$ is an ordered unit space with order unit the identity operator. Using the identification $M_n(B(\H))=B(\H^n)$ we let $M_n(B(H))_+=B(H^n)_+$. So we see that $B(\H)$ is an operator system.

Let $L$ be an operator system. Then any subspace $S\subset L$ that is selfadjoint, i.e. $S^*\subset S$, and contains the order unit of $L$ is again an operator system with the induced matrix order. So unital $C^*$-algebras and selfadjoint subspaces of unital $C^*$-algebras containig the identity are operator systems.

Note that a unital complete order isomorphism between unital $C^*$-algebras must be a $*$-isomorphism [Cho74, Corollary 3.2]. So unital $C^*$-algebras are completely characterized by their matrix order. They are not characterized by their order. For instance take the opposite algebra $A^{op}$ of a unital $C^*$-algebra $A$. Then $A_+=A_+^{op}$ but $A$ and $A^{op}$ are not $*$-isomorphic. Obviously $M_2(A)_+\not=M_2(A^{op})_+$.

Footnotes

... cone²⁰

A cone $K$ is a subset of a vector space, such that $K+K\subset K$ and ${\mathbb{R}}_+K\subset K$. If moreover $(-K)\cap K=\{0\}$ holds, $K$ is a proper cone.

... holds²¹

If $V^+=M_1(V)^+$ is a proper cone, by the matrix condition all the cones $M_n(V)^+$ will be proper.

... isomorphism²²

Note that some authors don't include surjectivity in the definition.