PlanetMath: ordering of self-adjoints

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ordering of self-adjoints

(Theorem)

Let $\mathcal{A}$ be a -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.

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.

Properties:

$a\leq b \;\Rightarrow\; c^*a\,c\leq c^*b\,c\;\;$ for every $c \in \mathcal{A}$ .
If and are invertible and $a \leq b$ , then $b^{-1} \leq a^{-1}$ .
If $\mathcal{A}$ has an identity element , then $-\Vert a\Vert e \leq a \leq \Vert a\Vert e\;$ for every $a \in \mathcal{A}_{sa}$ .
$-b \leq a \leq b \;\Rightarrow \;\Vert a\Vert \leq \Vert b\Vert$ .

Remark:

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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Cross-references: restricted, ordering relation, proof, identity element, invertible, ordered topological vector space, relation, partial order, self-adjoint elements, positive elements
There are 2 references to this entry.

This is version 4 of ordering of self-adjoints, born on 2007-08-28, modified 2007-08-28.
Object id is 9900, canonical name is OrderingOfSelfAdjoints.
Accessed 440 times total.

Classification:

AMS MSC:

46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras)

Pending Errata and Addenda

None.

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