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.

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.