${C}^{*}$algebras have approximate identities
In this entry $\le $ has three different meanings:

1.
 The ordering of selfadjoint elements (http://planetmath.org/OrderingOfSelfAdjoints) of a given ${C}^{*}$algebra (http://planetmath.org/CAlgebra).

2.
 The usual order (http://planetmath.org/PartialOrder) in $\mathbb{R}$.

3.
 The 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 ${({e}_{\lambda})}_{\lambda \in \mathrm{\Lambda}}$. Moreover, the approximate identity ${({e}_{\lambda})}_{\lambda \in \mathrm{\Lambda}}$ can be chosen to the following :

•
$0\le {e}_{\lambda}\mathit{\hspace{1em}}{\forall}_{\lambda \in \mathrm{\Lambda}}$

•
$\parallel {e}_{\lambda}\parallel \le 1\mathit{\hspace{1em}}{\forall}_{\lambda \in \mathrm{\Lambda}}$

•
$\lambda \le \mu \Rightarrow {e}_{\lambda}\le {e}_{\mu}$, i.e. ${({e}_{\lambda})}_{\lambda \in \mathrm{\Lambda}}$ is increasing.
For separable^{} (http://planetmath.org/Separable) ${C}^{*}$algebras the approximate identity can be chosen as an increasing sequence $0\le {e}_{1}\le {e}_{2}\le \mathrm{\dots}$ of normone elements.
Title  ${C}^{*}$algebras have approximate identities 

Canonical name  CalgebrasHaveApproximateIdentities 
Date of creation  20130322 17:30:40 
Last modified on  20130322 17:30:40 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  4 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 46L05 