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

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

1. 1.

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

2. 2.

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

3. 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.

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 the following :

• $0\leq e_{\lambda}\;\;\;\;\forall_{\lambda\in\Lambda}$

• $\|e_{\lambda}\|\leq 1\;\;\;\;\forall_{\lambda\in\Lambda}$

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

For separable (http://planetmath.org/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.

Title $C^{*}$-algebras have approximate identities CalgebrasHaveApproximateIdentities 2013-03-22 17:30:40 2013-03-22 17:30:40 asteroid (17536) asteroid (17536) 4 asteroid (17536) Theorem msc 46L05