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[parent] $C^*$-algebras have approximate identities (Theorem)

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

  1. - The ordering of self-adjoint elements of a given $C^*$ -algebra.
  2. - The usual order in $\mathbb{R}$ .
  3. - 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 $C^*$ -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:

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

For separable $C^*$ -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, theorem, 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 1240 times total.

Classification:
AMS MSC46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras)

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