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Let $\mathcal{A}$ be a $C^*$ -algebra. Let $\mathcal{A}^+$ denote the set of positive elements of $\mathcal{A}$ and $\mathcal{A}_{sa}$ denote the set of self-adjoint elements of $\mathcal{A}$ .
Since $\mathcal{A}^+$ is a proper convex cone (see this entry), we can define a partial order $\leq$ on the set $\mathcal{A}_{sa}$ , by setting
$a\leq b$ if and only if $b-a \in \mathcal{A}^+$ , i.e. $b-a$ is positive.
Theorem - The relation $\leq$ is a partial order relation on $\mathcal{A}_{sa}$ . Moreover, $\leq$ turns $\mathcal{A}_{sa}$ into an ordered topological vector space.
- $a\leq b \;\Rightarrow\; c^*a\,c\leq c^*b\,c\;\;$ for every $c \in \mathcal{A}$ .
- If $a$ and $b$ are invertible and $a \leq b$ , then $b^{-1} \leq a^{-1}$ .
- If $\mathcal{A}$ has an identity element $e$ , then $-\|a\|e \leq a \leq \|a\|e\;$ for every $a \in \mathcal{A}_{sa}$ .
- $-b \leq a \leq b \;\Rightarrow \;\|a\| \leq \|b\|$ .
The proof that $\leq$ is partial order makes no use of the self-adjointness property. In fact, $\mathcal{A}$ itself is an ordered topological vector space under the relation $\leq$ .
However, it turns out that this ordering relation provides its most usefulness when restricted to self-adjoint elements. For example, some of the above properties would not hold if we did not restrict to $\mathcal{A}_{sa}$ .
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