Definition 41.11.9 (Maximal Operator System Structure).label Let $S$ be an operator system with order unit $e$. For each $n \in \natp$, $C_{n} = \ol{M_n(\complex)_+ \otimes S_+}\subset M_{n}(S)$, then $\seq{C_n}$ defines a matrix order on $S$, and $(S, \seq{C_n}, e)$ is an operator system. The triple $\text{OMAX}(S) = (S, \seq{C_n}, e)$ is the maximal operator system structure on $S$.