Editing C*-algebra (section)

=== Quotients and approximate identities ===

Any C*-algebra ''A'' has an [[approximate identity]].  In fact, there is a directed family {''e''<sub>λ</sub>}<sub>λ∈I</sub> of self-adjoint elements of ''A'' such that

:: <math> x e_\lambda \rightarrow x </math>
:: <math> 0 \leq e_\lambda \leq e_\mu \leq 1\quad \mbox{ whenever } \lambda \leq \mu. </math>

: In case ''A'' is separable, ''A'' has a sequential approximate identity. More generally, ''A'' will have a sequential approximate identity if and only if ''A'' contains a '''[[Hereditary C*-subalgebra|strictly positive element]]''', i.e. a positive element ''h'' such that ''hAh'' is dense in ''A''.

Using approximate identities, one can show that the algebraic [[quotient]] of a C*-algebra by a closed proper two-sided [[Ideal (ring theory)|ideal]], with the natural norm, is a C*-algebra.

Similarly, a closed two-sided ideal of a C*-algebra is itself a C*-algebra.