Let $\mathcal{A}$ be a $C^{*}$-algebra (http://planetmath.org/CAlgebra). 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 (http://planetmath.org/Cone5) (see this entry (http://planetmath.org/PositiveElement3)), 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.

The relation $\leq$ is a partial order relation on $\mathcal{A}_{sa}$. Moreover, $\leq$ turns $\mathcal{A}_{sa}$ into an ordered topological vector space.

## 0.0.1 Properties:

• $a\leq b\;\Rightarrow\;c^{*}a\,c\leq c^{*}b\,c\;\;$ 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 $a\in\mathcal{A}_{sa}$.

• $-b\leq a\leq b\;\Rightarrow\;\|a\|\leq\|b\|$.

## 0.0.2 Remark:

The proof that $\leq$ is partial order makes no use of the self-adjointness . 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 would not hold if we did not restrict to $\mathcal{A}_{sa}$.

Title ordering of self-adjoints OrderingOfSelfadjoints 2013-03-22 17:30:37 2013-03-22 17:30:37 asteroid (17536) asteroid (17536) 7 asteroid (17536) Theorem msc 46L05