Operator system

Remarks

Example: in the C*-algebra $C(\mathbb{T})$ of continuous functions on the unit circle $\mathbb{T} = \set{z \in \C : \abs{z} = 1}$, let $S$ be the span of $1, z, \bar z$. It contains $1$, and $(a + b z + c \bar z)^* = \bar a + \bar c z + \bar b \bar z$ lies in $S$, so $S$ is an operator system. Non-example: the span of $1$ and $z$ in $C(\mathbb{T})$ contains $1$ but not $\bar z$. Indeed, if $\bar z = a + b z$ on $\mathbb{T}$, then $z = 1$ and $z = -1$ give $a = 0$ and $b = 1$, but then $z = i$ gives $-i = i$. Every self-adjoint element $h$ of an operator system $S$ satisfies $-\norm{h} 1 \le h \le \norm{h} 1$, so $S$ is spanned by its positive elements. Many authors take $A = B(H)$ for a Hilbert space $H$, and some also require $S$ to be norm closed.