ordering of selfadjoints
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 selfadjoint 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^{} $\le $ on the set ${\mathcal{A}}_{sa}$, by setting
$a\le b$ if and only if $ba\in {\mathcal{A}}^{+}$, i.e. $ba$ is positive.
Theorem  The relation^{} $\le $ is a partial order relation on ${\mathcal{A}}_{sa}$. Moreover, $\le $ turns ${\mathcal{A}}_{sa}$ into an ordered topological vector space.
0.0.1 Properties:

•
$a\le b\Rightarrow {c}^{*}ac\le {c}^{*}bc$ for every $c\in \mathcal{A}$.

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If $a$ and $b$ are invertible and $a\le b$, then ${b}^{1}\le {a}^{1}$.

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If $\mathcal{A}$ has an identity element^{} $e$, then $\parallel a\parallel e\le a\le \parallel a\parallel e$ for every $a\in {\mathcal{A}}_{sa}$.

•
$b\le a\le b\Rightarrow \parallel a\parallel \le \parallel b\parallel $.
0.0.2 Remark:
The proof that $\le $ is partial order makes no use of the selfadjointness . In fact, $\mathcal{A}$ itself is an ordered topological vector space under the relation $\le $.
However, it turns out that this ordering relation provides its most usefulness when restricted to selfadjoint elements. For example, some of the above would not hold if we did not restrict to ${\mathcal{A}}_{sa}$.
Title  ordering of selfadjoints 

Canonical name  OrderingOfSelfadjoints 
Date of creation  20130322 17:30:37 
Last modified on  20130322 17:30:37 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  7 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 46L05 