Corollary 36.12.6.label Let $A$ be a $C^{*}$-algebra, then $A$ admits an increasing approximate identity.
Proof. Identify $A$ as a self-adjoint two-sided ideal of its unitisation $\td A$, which is a $C^{*}$-algebra by Proposition 36.12.1. Applying Theorem 36.12.5 to $A$ and $\bracs{x^*|x \in A}$ as left and right ideals, respectively, yields that the net constructed by the theorem is an increasing approximate identity for $A$.$\square$
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