Definition 36.12.3 (Increasing Approximate Identity).label Let $A$ be a $C^{*}$-algebra and $\angles{e_\beta}_{\beta \in B}\subset A$ be an approximate identity, then $\angles{e_\beta}_{\beta \in B}$ is increasing if:

1. (1)

   For each $\beta \in B$, $e_{\beta} \ge 0$.

2. (2)

   For each $\beta, \gamma \in B$ with $\beta \le \gamma$, $e_{\beta} \le e_{\gamma}$.