$C^{*}$-algebras have approximate identities

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

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

2. - The usual order (http://planetmath.org/PartialOrder) in $ℝ$.

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 norm-one elements.