Why do closed ideals of C*-algebras have approximate units

Question

In the Blackadar's book Operator algebras: theory of $C^\ast$-algebras and von Neumann algebras, on p. 103, there is a statement "approximate units for $J$", here $J$ is an ideal of a C*-algebra $A$, but I do not know why an ideal also has approximate units. We know that $A^+$ is the approximate units of $A$; does the intersection of $A^+$ and $J$ is the approximate units for $J$?

Answer 1

Closed ideals of C*-algebras are ${}^\ast$-closed hence they are C*-algebras. Every C*-algebra has a bounded approximate identity consisting of self-adjoint elements. The proof is contained in every book concerning operator algebras. An approximate identity gives you approximate units trivially.

For a good treatment of this business consult Davidson's book $C^\ast$-star algebras by examples (Theorems I.4.8 and I.9.16).