PlanetMath: $C^*$-algebras have approximate identities

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-algebras have approximate identities

(Theorem)

In this entry $\leq$ has three different meanings:

- The ordering of self-adjoint elements of a given -algebra.
- The usual order in $\mathbb{R}$ .
- The order of a directed set taken as the domain of a given net.

It will be clear from the context which one is being used.

Theorem - Every -algebra has an approximate identity $(e_{\lambda})_{\lambda \in \Lambda}$ . Moreover, the approximate identity $(e_{\lambda})_{\lambda \in \Lambda}$ can be chosen to satisfy the following properties:

$0\leq e_{\lambda}\;\;\;\;\forall_{\lambda \in \Lambda}$
$\Vert e_{\lambda}\Vert \leq 1\;\;\;\;\forall_{\lambda \in \Lambda}$
$\lambda \leq \mu\; \Rightarrow \;e_{\lambda}\leq e_{\mu}$ , i.e. $(e_{\lambda})_{\lambda \in \Lambda}$ is increasing.

For separable -algebras the approximate identity can be chosen as an increasing sequence $0\leq e_1 \leq e_2 \leq \dots$ of norm-one elements.

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Cross-references: sequence, increasing, approximate identity, clear, net, domain, directed set
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This is version 1 of $C^*$

-algebras have approximate identities, born on 2007-08-28.
Object id is 9901, canonical name is CAlgebrasHaveApproximateIdentities.
Accessed 680 times total.

Classification:

AMS MSC:

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

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